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The production of hypersonic shock waves in an electrothermal diaphragm shock tube Phillips, Malvern Gordon Rutherford 1969

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\\7b  THE PRODUCTION OF HYPERSONIC SHOCK WAVES IN AN ELECTROTHERMAL DIAPHRAGM SHOCK TUBE by  MALVERN GORDON RUTHERFORD PHILLIPS  B.Sc,  U n i v e r s i t y o f B r i t i s h Columbia, 1965  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of  PHYSICS  We a c c e p t t h i s t h e s i s as conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA July,  1969  In p r e s e n t i n g an the  thesis  advanced degree at Library  I further for  this  shall  the  his  of  this  agree that  written  University  of  permission  representatives. thesis  f u l f i l m e n t of  make i t f r e e l y  s c h o l a r l y p u r p o s e s may  by  in p a r t i a l  be  available for for extensive  g r a n t e d by  for financial  gain  of  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  the  It is understood  permission.  Department  British  Columbia  shall  requirements  Columbia,  Head o f my  be  I agree  r e f e r e n c e and copying of  that  not  the  that  Study.  this  thesis  Department  c o p y i n g or  for  or  publication  allowed without  my  ii ABSTRACT The o p e r a t i o n o f a diaphragm  shock tube o f 5 cm  inside  diameter i n which the d r i v e r gas i s heated by the d i s c h a r g e of e l e c t r i c a l energy ( ~  10  j o u l e s ) i s analyzed i n d e t a i l .  A  technique i s d e s c r i b e d f o r the measurement o f the heated d r i v e r gas p r e s s u r e and e m p i r i c a l r e l a t i o n s are o b t a i n e d which enable the  shock speed to be c a l c u l a t e d from a knowledge o f the d i s -  charge v o l t a g e and t e s t gas p r e s s u r e . initially  Using h e l i u m d r i v e r gas  at atmospheric p r e s s u r e , shock Mach numbers o f about  20 are o b t a i n e d i n argon at an i n i t i a l p r e s s u r e o f about 1 Torr.  The s e p a r a t i o n o f shock f r o n t and c o n t a c t s u r f a c e i s  analyzed by means o f a convenient shock r e f l e c t i o n technique u s i n g a smear camera. are  The p r o p e r t i e s o f the. shock-heated gas  shown to agree w i t h the p r e d i c t i o n s o f s t a n d a r d shock wave  t h e o r y , which y i e l d s a temperature o f about 1.3*10^°K and an 17 e l e c t r o n d e n s i t y o f about IQ shock i n argon at 0.5  Torr.  -3 cm  In t h i s  f o r the case o f a Mach 20 case the shock-heated gas  sample i s observed to be about 5 cm i n l e n g t h at a p o s i t i o n 1.2  meters from the  diaphragm.  iii TABLE OF  CONTENTS  Page ABSTRACT  i i  TABLE OF CONTENTS  i i i  L I S T OF TABLES  v  L I S T OF FIGURES  vi  ACKNOWLEDGEMENTS  ix  CHAPTER.l'  INTRODUCTION  1.1  Instructions  1.2  The B a s i c  1.3  Outline  CHAPTER  2  t o the Reader  1  Problem  1  o f the Thesis  SHOCK WAVES AND  6  SHOCK TUBES  2.1  Introduction  2.2  Shock Waves  i n I d e a l Gases  2.3  Shock Waves  i n Real  2.4  Dependence o f Shock Speed on D i a p h r a g m Pressure Ratio  15  2.5  S o l u t i o n of the Equations  19  2.6  The S h o c k - H e a t e d Gas  25  2.7  Boundary  28  2.8  The E f f e c t  CHAPTER 3  9  Monatomic  9 Gases  Layers o f Shock Tube E x t r e m i t i e s  13  29  APPARATUS  3.1  Introduction  31  3.2  The D r i v e r  31  3.3  The Dischai-ge  3.4  The D i a p h r a g m  Section Circuit  34 34  iv 3.5  The Test S e c t i o n  3.6  Instrumentation  41  3.7  E l e c t r i c a l Measurements  43  CHAPTER 4  '  38  THE DRIVER  4.1  Introduction  45  4.2  The D r i v i n g Mechanism  45  4.3  D e t e r m i n a t i o n o f the D r i v e r Gas P r e s s u r e  49  CHAPTER 5  COMPARISON OF SHOCK TUBE OPERATION WITH IDEAL THEORY  5.1  Introduction  62  5.2  Dependence o f Shock Speed on A x i a l P o s i t i o n  63  5.3  Comparison o f Shock Tube O p e r a t i o n w i t h I d e a l Theory  67  5.4  The E f f i c i e n c y o f the Shock Tube  77  CHAPTER 6  THE SHOCK-HEATED GAS  6.1  Introduction  81  6.2  E x t e n t o f the Region o f Shock-Heated Gas  83  6.3  C u r r e n t s i n the Shock-Heated Gas  105  6.4  P r o p e r t i e s o f the Shock-Heated Gas  108  CHAPTER 7  CONCLUSIONS  7.1  Conclusions  7.2  Suggestions  119 f o r Future Work  BIBLIOGRAPHY APPENDIX - PRESSURE PROBES  122 1  124 127  V  L I S T OF  TABLES  Page Table  I  S h o c k Tube D r i v e r  Table  II  Electrical  Table  III  Thermal E f f i c i e n c y  Conditions  Efficiency  72  o f the Shock  o f the Shock  Tube  Tube  79 80  vi LIST OF FIGURES  Page 1-1  P r e s s u r e - D r i v e n Shock Tube  3  1- 2  Comparison of Types o f Shock Tubes  7  2- 1  Shock Wave i n L a b o r a t o r y and S h o c k - F i x e d  10  Frames o f R e f e r e n c e 2-2  Gas Flow i n Shock Tube A f t e r Diaphragm B u r s t s  18  2-3 2-4  P r e s s u r e R a t i o a c r o s s Shocks i n Argon D e n s i t y and V e l o c i t y R a t i o s a c r o s s Shocks i n Argon  20 21  2-5  Temperature B e h i n d Shocks i n Argon  22  2-6  Degree of I o n i z a t i o n (a) i n Shock-Heated Argon  23  2-7  Shock Mach Number (M) as. a F u n c t i o n o f Diaphragm P r e s s u r e R a t i o (p^/p-^) f ° Shocks i n Argon D r i v e n by H e l i u m  24  r  2- 8  L u m i n o s i t y S t r u c t u r e i n S t r o n g Shock  3- 1  Schematic Diagram of V e r t i c a l S e c t i o n t h r o u g h  .27 32  Shock Tube A x i s 3-2  D r i v e r S e c t i o n o f Shock Tube  33  3-3  D e t a i l s of Discharge C i r c u i t  35  3-4  Two Views o f the D r i v e r S e c t i o n  37  3-5 3-6  Diaphragms B e f o r e and A f t e r B u r s t i n g H o r i z o n t a l A x i a l S e c t i o n through S p e c i a l Test S e c t i o n  37 40  3- 7  S i m p l i f i e d Diagram o f P r e s s u r e Probe and Housing  42  4- 1  Shock Speeds W i t h and W i t h o u t B a c k s t r a p  47  4-2  Dependence o f Shock Speed on D i s c h a r g e C u r r e n t For D r i v e r w i t h B a c k s t r a p  48  4-3  Diaphragm P e t a l Geometry  50  Vll.  4-4  Arrangement f o r Observing  4-5  Smear P h o t o g r a p h o f D i a p h r a g m O p e n i n g  4-6  Dependence o f D i a p h r a g m O p e n i n g Time Bank V o l t a g e  on  53  4-7 ,v  Dependence o f D i a p h r a g m O p e n i n g Time Diaphragm T h i c k n e s s  on  54  4-8  Arrangement f o r O b s e r v i n g Diaphragm Opening with Cold Pressurized Driver  56  4-9  P h o t o m u l t i p l i e r and P r e s s u r e T r a n s d u c e r Records o f Diaphragm Opening  57  4-10  D i a p h r a g m Dynamics  57  4-11  C u r r e n t Waveform and D i a p h r a g m R u p t u r e Time  60  4- 12  Times C h a r a c t e r i s t i c  60  5- 1  Dependence o f Shock Speed on D i s t a n c e the Diaphragm  from  64-65  5-2  Dependence o f Shock Speed on P o s i t i o n D i f f e r e n t Diaphragm T h i c k n e s s e s  for  69  5-3  Dependence  :  Diaphragm Opening  51 51  f o r C o l d D r i v e r Gas  of Discharge  o f L on t h e P r o d u c t  and D i a p h r a g m  v  • t. max op Dependence o f Shock Speed on P o s i t i o n f o r Different Driver Conditions  70  1  5-4  71  5-5  M e a s u r e d Shock Speed v e r s u s  log-j^Cp^/p,)(V =14KV)  73  5-6  M e a s u r e d Shock S p e e d v e r s u s  log-j^Cp^/p,)(V =10KV)  74  5-7  M e a s u r e d Shock Speed v e r s u s  log  75  5- 8  Initial  6- 1  Axial  6-2  Smear P i c t u r e Perpendicular  6-3  Smear P i c t u r e s  6-4  E x p l a n a t i o n o f the F e a t u r e s P h o t o g r a p h s o f F i g u r e 6-3  6-5  Dependence o f Shock Mach Number (M) and ShockH e a t e d Gas T h i c k n e s s (1 ) on T e s t Gas P r e s s u r e  D r i v e r Gas  c  c  1 0  ( p ^ / P ] ^ (V = 7KV)  Flow P a t t e r n  S e c t i o n o f Shock Tube Showing  76 R e g i o n s o f Flow  o f Shock T a k e n w i t h S l i t t o Shock Tube A x i s o f Gas  82 82  Flow i n Shock Tube  87-89  o f t h e Smear Camera  90  92 (p,)  viii 6-6 6-7  Comparison o f Measured T h i c k n e s s o f ShockHeated Gas (1 ) w i t h I d e a l Value (1., ) m th/ C o n t a c t S u r f a c e Speed as a F u n c t i o n of Shock Speed  93 95  6-8  Comparison o f D e n s i t y R a t i o s  96  6-9  E x p e r i m e n t a l Check on t h e V a l i d i t y of E q u a t i o n (6.10)  100  6-10  Two S i t u a t i o n s i n w h i c h the I n t e g r a l o f E q u a t i o n (6.8) may be Approximated as i n E q u a t i o n (6.10)  101  6-11  Attempts t o Observe C u r r e n t s i n Shock-Heated Gas  107  6-12  Dependence of P r e s s u r e Probe S i g n a l A m p l i t u d e  110  v  on  P l  V  2 s  6-13  D e t a i l s o f Spark Gap Used t o Produce "Sound" P u l s e  112  6-14  Arrangement f o r M e a s u r i n g Sound Speed i n ShockHeated Gas  114  6-15  D e t e r m i n a t i o n o f Sound Speed i n Shock-Heated Gas  116  A-l  B a s i c F e a t u r e s o f P i e z o e l e c t r i c P r e s s u r e Probe  127  A-2  C o n s t r u c t i o n D e t a i l s of P r e s s u r e Probe Housing  132  A-3  T y p i c a l Pressure Probe"Signals  133  ix ACKNOWLEDGEMENTS I wish to thank Dr. F. L. Curzon f o r h i s p a t i e n t s u p e r v i s i o n and encouragement during the course research. Barnard,  The  suggestions  of t h i s  of Dr. B. A h l b o r n , Dr. A. J .  and Dr. J . Meyer i n the p r e p a r a t i o n o f the t h e s i s  are v e r y much a p p r e c i a t e d . I am  a l s o indebted  to a l l members o f the Plasma P h y s i c s  Group, p a r t i c u l a r l y Dr. B. A h l b o r n , Mr. J . D. Mr.  J . -P. Huni, and  Strachan, f o r h e l p f u l d i s c u s s i o n s .  I am g r a t e f u l to  Huni f o r the l o a n of the smear camera. The  staff  t e c h n i c a l a s s i s t a n c e o f Mr. A. F r a s e r and h i s  - i n p a r t i c u l a r , Mr.  and Mr.  T. Knopp, who  P. Haas, Mr.  D.  a l s o due to Mr.  to Mr.  R. Haines,  J . Lees and Mr.  w i t h the glassware, J . Aazam, Mr.  Stonebridge,  c o n s t r u c t e d v a r i o u s p a r t s of the  shock tube - i s g r a t e f u l l y acknowledged.  Mr.  Mr.  and  My  thanks are  f o r h i s help i n the student  shop,  E. W i l l i a m s , f o r t h e i r advice and to Mr.  J . Dooyeweerd, Mr.  R. Da Costa, and Mr. W.  D.  Sieberg,  Ratzlaff, for  c h e e r f u l a s s i s t a n c e i n the maintenance o f the equipment. F i n a l l y , I am very g r a t e f u l t o the N a t i o n a l Research C o u n c i l of Canada f o r f i n a n c i a l a s s i s t a n c e throughout course  o f the work.  help  the  CHAPTER 1  1.1  INTRODUCTION  I n s t r u c t i o n s to the Reader T h i s t h e s i s d e a l s w i t h the l a b o r a t o r y p r o d u c t i o n o f  s t r o n g shock waves by e l e c t r i c a l l y h e a t i n g the d r i v e r gas i n a diaphragm shock tube.  S e c t i o n 1.2 s e t s out t h e b a s i c concepts  r e l e v a n t t o p r e s e n t shock tube t e c h n o l o g y , d i s c u s s e s the l i m i t a t i o n s on c o n v e n t i o n a l shock tubes, and d e s c r i b e s a new shock tube (invented by P. R. Smy^ ' ^) which i s the s u b j e c t o f 5  t h i s work.  6  Readers thoroughly f a m i l i a r with the f i e l d are  r e f e r r e d f i r s t t o S e c t i o n 1.3, f o r an o u t l i n e o f the t h e s i s , and then t o Chapter 7, where the c o n c l u s i o n s and c o n t r i b u t i o n s are  summarized.  1.2  The B a s i c Problem When a shock wave passes through a gas, the gas i s com-  p r e s s e d and heated.  From a knowledge o f the shock speed, V  and the i n i t i a l gas p r e s s u r e , d e n s i t y , and temperature (p,, p,, and T,, r e s p e c t i v e l y ) , the p r o p e r t i e s o f the compressed gas (P2> P £ » "^2^  c  accuracy.  a  n  ^  e  c a  l  c u  l  a - t e  d»  i d e a l l y at l e a s t , w i t h good  I f the shock speed i s s u f f i c i e n t l y h i g h , the heated  gas may be i o n i z e d .  The a b i l i t y of shock waves t o produce hot  gas samples w i t h p r e d i c t a b l e p r o p e r t i e s has i n r e c e n t years aroused the i n t e r e s t o f s c i e n t i s t s and e n g i n e e r s . P a r t i c u l a r l y , i n the general f i e l d  o f plasma p h y s i c s ,  shock waves have been and continue t o be very u s e f u l t o o l s . is  important t o check, f o r example, t h e o r i e s f o r plasma  It  t r a n s p o r t p r o p e r t i e s under known c o n d i t i o n s . f o r atomic t r a n s i t i o n p r o b a b i l i t i e s  Accurate values  are n e c e s s a r y i n a s t r o -  p h y s i c a l s t u d i e s as w e l l as i n the area of s p e c t r o s c o p i c d i a g n o s i s o f l a b o r a t o r y plasmas.  Shock tubes have proved u s e f u l  i n determining these q u a n t i t i e s .  Theories f o r s p e c t r a l  broadening and s h i f t  are c u r r e n t l y being complemented  o b t a i n e d from shock wave experiments.  line  by data  The mechanisms through  which i o n i z a t i o n o c c u r s , though not y e t w e l l understood, have been p a r t i a l l y e l u c i d a t e d i n work w i t h shock waves. In the area o f magnetohydrodynamics,  shock wave plasmas  are v e r y u s e f u l because the plasma can be produced i n the absence o f magnetic and e l e c t r i c f i e l d s .  The e x p e r i m e n t a l i s t  i s t h e r e f o r e at l i b e r t y t o impose f i e l d s o f h i s c h o i c e . Two techniques f o r p r o d u c i n g shock waves i n the l a b o r a t o r y have become c o n v e n t i o n a l .  Each has advantageous  aspects and l i m i t a t i o n s . The p r e s s u r e d r i v e n shock tube, shown s c h e m a t i c a l l y i n F i g u r e 1-1, c o n s i s t s o f a long tube d i v i d e d i n t o two s e c t i o n s by a t h i n membrane.  The d r i v e r s e c t i o n i s f i l l e d w i t h gas at  a h i g h p r e s s u r e , p^, w h i l e the t e s t s e c t i o n c o n t a i n s the gas to be s t u d i e d a t a s u i t a b l e p r e s s u r e , p^.  When the membrane  i s broken, d r i v e r gas rushes i n t o the t e s t gas at h i g h speed, Vj.  A shock wave then develops and propagates ahead o f the  d r i v e r gas " p i s t o n " a t a speed, *V", w h i l e the compressed s  test  gas c o l l e c t s between the shock f r o n t and the l e a d i n g edge of the d r i v e r gas, which i s c a l l e d the c o n t a c t s u r f a c e 1-lb).  (Figure  The f e a t u r e s o f the gas flow i n the shock tube are  3  D r i v e r Gas  T e s t Gas  P r e s s u r e P4  P r e s s u r e p^  Driver Section  Diaphragm  Test S e c t i o n  (a) B e f o r e Diaphragm B u r s t s  Compressed T e s t Gas  T e s t Gas  J Expanded 1 Driver l Gas 1 1 •  Contact Surface  Shock Front  (b) Time t ^ A f t e r Diaphragm B u r s t s t  £  x  (c) x - t Diagram  F i g u r e 1-1  P r e s s u r e - D r i v e n Shock Tube  c o n v e n i e n t l y r e p r e s e n t e d i n an x - t diagram  (Figure 1 - l c ) .  As  the shock proceeds a l o n g the tube, the sample o f shock-heated gas i n c r e a s e s i n l e n g t h .  Shock tubes of t h i s type have the  f a v o r a b l e a t t r i b u t e s o f p r o d u c i n g shock waves o f almost c o n s t a n t speed and r e l a t i v e l y l a r g e samples properties. may be used.  o f hot gas w i t h p r e d i c t a b l e  A l s o , any combination of t e s t and d r i v e r  gases  However, the shock waves cannot be launched  r e p r o d u c i b l y i n time due t o the mechanical u n c e r t a i n t i e s a s s o c i a t e d w i t h the deformation and r u p t u r e of the diaphragm. Furthermore, the a p p l i c a t i o n o f t h i s type o f shock tube i s s e v e r e l y l i m i t e d by the f a c t that even f o r an i n f i n i t e p r e s s u r e r a t i o , p^/p-^, the shock Mach number  ( r a t i o o f shock  speed t o sound speed i n the c o l d t e s t gas) cannot  where  ^  and tfjj. are the a d i a b a t i c exponents  initial  exceed  of t e s t and d r i v e r  gases, r e s p e c t i v e l y , w h i l e a^ and a^ are the sound speeds.  For  h e l i u m d r i v e r gas and argon t e s t gas, f o r example, the maximum Mach number i s about 12.5 but t e c h n i c a l r e s t r i c t i o n s  on V^/]?i  p r a c t i c a l l y l i m i t t h i s v a l u e t o about 10, r e s u l t i n g i n only about 1% i o n i z a t i o n .  To achieve a h i g h e r degree o f i o n i z a t i o n ,  f a s t e r shock waves are r e q u i r e d .  For t h i s purpose, equation (1.1)  suggests t h a t , f o r a g i v e n t e s t gas, one should use a d r i v e r gas of h i g h sound speed, a^.  The sound speed o f a gas i s proper -  1/2 t i o n a l to (T/m) ' , where T i s the absolute temperature o f the gas, and m i s i t s m o l e c u l a r weight.  Because  of i t s low m,  hydrogen d r i v e r gas p r o v i d e s the f a s t e s t shocks.  However, i t s  5 high c o m b u s t i b i l i t y requires that s t r i c t  safety precautions  observed, p a r t i c u l a r l y i f an attempt i s to be made to a^ by r a i s i n g the d r i v e r gas  temperature.  increase  Helium i s the next  b e s t c h o i c e , from the p o i n t of view of molecular  weight,  because i t i s i n e r t , presents  no hazard when heated.  d r i v e r gas  i n c r e a s e d most e f f i c i e n t l y  heating  sound speed can be  the gas w i t h  an e l e c t r i c a l  be  and,  The by  discharge. f53")  E a r l y e f f o r t s to e l e c t r i c a l l y heat the d r i v e r gas^ employed a p a i r of e l e c t r o d e s at one a uniform the gas  pressure,  p,.  end  J  o f a tube i n i t i a l l y  at  Discharge of a c a p a c i t o r bank through  between the e l e c t r o d e s c r e a t e s  which expands t o d r i v e a shock wave.  a volume of hot I t was  gas  quickly realized  t h a t the shock speeds c o u l d be. f u r t h e r i n c r e a s e d by e l e c t r o magnetically gas  p r o j e c t i n g the hot d r i v e r gas  To p r o v i d e  the e l e c t r o m a g n e t i c  leads are arranged i n the "backstrap"  i n t o the c o l d t e s t  f o r c e , the  c o n f i g u r a t i o n (Figure l-2b) .  Then, the i n t e r a c t i o n between the gaseous discharge the magnetic f i e l d  from the backstrap  and-these shock tubes can be two  current  can be  down the  achieved  tube.  in this  triggered reproducibly.  very s e r i o u s l i m i t a t i o n s a r i s e :  and  current r e s u l t s i n a  s t r o n g J x B f o r c e p r o p e l l i n g the d r i v e r gas Shock Mach numbers i n excess of 100  current  way,  However,  the shock r a p i d l y  d e c e l e r a t e s , so t h a t experiments must be performed c l o s e to the d r i v e r where only a s m a l l amount of shock-heated gas accumulated behind the shock f r o n t ; shock-heated gas  cannot always be  the p r o p e r t i e s of  has  the  a c c u r a t e l y p r e d i c t e d from  the shock speed s i n c e the discharge  current often p e r s i s t s i n  6 the shock-heated t e s t gas and s i n c e t u r b u l e n t mixing of t e s t and d r i v e r gases i s o f t e n so complete fx  that no separated shock  41  f r o n t i s observed^ * . 1  Smy^combined  both types of shock tubes by  an e l e c t r o m a g n e t i c d r i v e r arrangement w i t h the hope t h a t the advantages (Figure l-2c) . obtained^ .  i n a diaphragm  employing  shock tube  of both types would be  combined  High Mach numbers and low a t t e n u a t i o n were indeed  A l s o , the shock tube can be c o n v e n i e n t l y  triggered.  However, i t i s not a p r i o r i c l e a r t h a t such a h y b r i d shock tube i n h e r i t s a l l the f i n e q u a l i t i e s o f both p a r e n t s and none of t h e i r defects.  From an o p e r a t i o n a l p o i n t of view, there i s one  important q u e s t i o n :  Is a s i z e a b l e volume of shock-heated  allgas  produced, and can i t s p r o p e r t i e s be p r e d i c t e d from the measured shock speed?  S e v e r a l s u b s i d i a r y q u e s t i o n s a l s o come to mind:  What i s the mechanism r e s p o n s i b l e f o r p r o d u c i n g the h i g h speed shocks?  What r o l e does the-diaphragm p l a y ?  Can the shock speed  be p r e d i c t e d from a knowledge o f the e l e c t r i c a l energy s t o r e d i n the c a p a c i t o r bank and from the i n i t i a l p r e s s u r e s i n the t e s t and d r i v e r s e c t i o n s ?  Some of these q u e s t i o n s are i n t e r - r e l a t e d  but a l l must be answered  i f a thorough u n d e r s t a n d i n g o f the  shock tube i s t o be o b t a i n e d .  I t was  the purpose of the work  r e p o r t e d i n t h i s t h e s i s to seek answers to these q u e s t i o n s w i t h the hope o f shedding l i g h t on the flow the Smy 1.3  processes occurring i n  shock tube.  O u t l i n e o f the T h e s i s In Chapter 2 we  review the standard shock tube theory  7  D r i v e r Gas P r e s s u r e P4  Driver Section  T e s t Gas P r e s s u r e pi  Diaphragm  Test Section  (a) C o n v e n t i o n a l P r e s s u r e - D r i v e n Shock Tube  Backstrap FO  ^Current. Net F o r c e '^Backstrap Magnetic  T e s t Gas Pressure p Field  Capacitor  HP (b) E l e c t r o m a g n e t i c a l l y D r i v e n Shock Tube D r i v e r Gas I n i t i a l l y a t Atmospheric T e s t Gas  Electrodes  Pressure p Diaphragm  HH  Capacitor (c) Sir.y's Shock Tube F i g u r e 1-2  Comparison of Types of Shock Tubes  Pressure  p e r t i n e n t t o our experiments.  In p a r t i c u l a r , , the equations  r e l a t i n g the shock-heated gas c o n d i t i o n s to the i n i t i a l  test  gas c o n d i t i o n s and the shock speed are d i s c u s s e d , as w e l l as the  dependence of the shock speed on the i n i t i a l  t e s t gas c o n d i t i o n s i n a c o n v e n t i o n a l diaphragm  driver  and  shock tube.  We  p r e s e n t r e s u l t s o f c a l c u l a t i o n s f o r the i n t e r e s t i n g case o f shocks i n argon d r i v e n by helium.  The c o n s t r u c t i o n o f the  Smy  shock tube i s d i s c u s s e d i n Chapter 3 together w i t h the b a s i c i n s t r u m e n t a t i o n employed  i n the i n v e s t i g a t i o n .  Some improvements  which we have made to the design of the shock tube are noted i n S e c t i o n s 3.2  and 3.4.  Our i n v e s t i g a t i o n of the d r i v i n g mechanism  r e s p o n s i b l e f o r the p r o d u c t i o n of the h i g h speed shocks i s p r e s e n t e d i n Chapter 4 where the dominance o f the e l e c t r o t h e r m a l d r i v e r h e a t i n g over e l e c t r o m a g n e t i c f o r c e s i s e s t a b l i s h e d . S e c t i o n 4.3,  a technique which we have developed to measure the  d r i v e r pressure i s described.  An e m p i r i c a l r e l a t i o n f o r pre-  d i c t i n g the shock speed from the known i n i t i a l  conditions  found f o r t h i s tube and i s d i s c u s s e d i n Chapter 5. the  In  shock-heated gas i t s e l f i s examined.  We  was  In Chapter  6,  applied a reflected  shock technique to i n v e s t i g a t e the d u r a t i o n of u n i f o r m flow and the  mixing of t e s t and d r i v e r gases.  The  thermodynamic  p r o p e r t i e s of the shock-heated gas were s t u d i e d by means o f p r e s s u r e and sound speed measurements. In  summary, we have used s e v e r a l e x p e r i m e n t a l techniques  and t h e o r e t i c a l models not p r e v i o u s l y a p p l i e d to i n v e s t i g a t i o n s of  t h i s nature to e s t a b l i s h that h i g h speed shock waves and w e l l -  behaved  plasmas  can be produced w i t h the Smy  shock tube.  9 CHAPTER 2  2.1  SHOCK WAVES AND SHOCK TUBES  Introduction The  purpose o f t h i s chapter  i s to present  the equations  r e l a t i n g the s t a t e o f the shock heated gas to the shock speed and  the i n i t i a l t e s t gas c o n d i t i o n s .  speed on the diaphragm pressure shock tube i s a l s o d i s c u s s e d .  The dependence o f shock  r a t i o , p^/p^, i n a c o n v e n t i o n a l The treatment i s p a r t i c u l a r i z e d  to the case o f shocks i n argon d r i v e n by helium. equations r e l a t i n g  the v a r i a b l e s on o p p o s i t e  f r o n t are d i s c u s s e d by many authors Therefore  2.2  The c l a s s i c a l  s i d e s o f the shock  (e.g. r e f e r e n c e s  t h e i r d e r i v a t i o n w i l l not be given  7 and 8 ) .  in detail  here.  Shock Waves i n I d e a l Gases Consider  a shock wave moving to the r i g h t  i n t o gas at r e s t (Figure 2 - l a ) .  at speed V  g  P r o p e r t i e s o f t h i s gas w i l l be  denoted by a s u b s c r i p t 1, while p r o p e r t i e s o f the gas behind the shock f r o n t , the shock heated gas, w i l l be denoted by a s u b s c r i p t 2.  We s h a l l assume t h a t a steady s t a t e e x i s t s and that through-  out each r e g i o n the gas i s i n a s t a t e o f thermodynamic equilibrium.  The gas flow i s assumed to be one-dimensional so  t h a t boundary l a y e r s formed a t the w a l l s o f t h e . v e s s e l the gas are ignored.. conduction,  confining  We a l s o n e g l e c t energy l o s s e s through  r a d i a t i o n , and c o n v e c t i o n .  In a frame o f r e f e r e n c e  i n which the shock wave i s a t r e s t (Figure 2-lb) , the flow velocities  are denoted by the symbol u.  conservation  The g e n e r a l  laws f o r mass, momentum, and energy r e q u i r e  that  10  P» h  '  2  T  P i , y. ,  2  •>  -fr v,  X  V  Undisturbed T e s t Gas  Shock Front  Shock-Heated T e s t Gas  T  (a) L a b o r a t o r y Frame o f Reference  P2, ?a »  u  2  =  V  2  T 2  •P.  ~~  " s  Shock-Heated T e s t Gas  f  x  u  V  Shock Front  l  i  >  T  l  = " s V  Undisturbed T e s t Gas  (b) S h o c k - F i x e d Frame c f R e f e r e n c e  hock Wave i n L a b o r a t o r y and Shock-Fixed Frames of R e f e r e n c e  =  J^U, ft  P z ^  + ^U,- =  +  7  Vi, -K-iu,  =  1  (2.1)  fX  (2.2)  h> + i"Uj  (2.3)  where j>, -p, and h denote mass d e n s i t y , p r e s s u r e , and s p e c i f i c enthalpy, r e s p e c t i v e l y .  Enthalpy i s d e f i n e d by  • •. h =  e  +  '  where e i s the i n t e r n a l energy per u n i t mass.  Equations  (2.4) (2.1) t o  (2.3) r e l a t e the gas c o n d i t i o n s behind the shock to those i n f r o n t of the shock.  They are sometimes r e f e r r e d to as the  Rankine-Hugoniot equations.  These equations may be w r i t t e n i n  a l a b o r a t o r y - f i x e d frame o f r e f e r e n c e by the t r a n s f o r m a t i o n u  l  u where v  2  " s  =  (2.5)  V  - v  2  - V  denotes the gas flow speed  2  In order t o determine  (2.6)  s  i n the l a b o r a t o r y .  the p r o p e r t i e s of r e g i o n 2, the  e q u a t i o n of s t a t e of the gas must be known as w e l l as the dependence of e on the other v a r i a b l e s . f  '  e  ~  =  For a p e r f e c t gas (2.7)  f T ( T  i  KL  where T i s absolute temperature,  -  1 ±  (2.8)  R i s the gas constant per u n i t  mass, k i s Boltzmann's constant and m i s the mass of a gas particle.  For a gas w i t h a constant s p e c i f i c heat r a t i o ,  $  ,  the enthalpy may be expressed as  Equations  (2.1) to (2.3) then  A  - Ji- -  yield  (*>-') ft  + U + Ok.  (2.10)  12 and  ^  -  ft  ~  (er. + op,  " ^ ' - ' ^  (2 11)  - U.-OP,  (  2  ,  1  1  }  f o r the d e n s i t y r a t i o and p r e s s u r e r a t i o across the shock. I t i s customary to i n t r o d u c e the Mach number o f the shock, M » V /a s  (2.12)  1  where a^ i s the sound speed i n the u n d i s t u r b e d gas of r e g i o n 1. For a p e r f e c t gas  One then o b t a i n s  ?r • j i  -  <- )  *, +1  2  VU.M*-y, +  i)(M (^.+i) a  +0  14  (2.i6)  A shock i s r e f e r r e d to as s t r o n g i f M (or e q u i v a l e n t l y , P2/P1) is  large.  I t i s i n t e r e s t i n g t o note t h a t as the shock s t r e n g t h  i n c r e a s e s , the d e n s i t y r a t i o approaches a l i m i t i n g v a l u e pf ( tfi + l ) / (  - 1) f o r a p e r f e c t gas.  I t should be p o i n t e d out t h a t the v a l i d i t y o f equations (2.1) to (2.3) does n o t depend on the i n t e r n a l s t r u c t u r e o f the shock wave p r o v i d e d t h a t t h i s s t r u c t u r e i s independent o f time. The shock wave i n f a c t c o n s i s t s o f a t r a n s i t i o n e x t e n t over which the gas p r o p e r t i e s change rapidly.  zone o f f i n i t e  c o n t i n u o u s l y , but  The width o f the zone turns out t o be o f the order  of magnitude  o f a few i n t e r p a r t i c l e d i s t a n c e s r e l a t i v e to the  gas i n s t a t e 1.  Throughout t h i s  zone, the e f f e c t s o f heat  c o n d u c t i o n and v i s c o s i t y cannot be ignored because of the l a r g e v e l o c i t y and temperature g r a d i e n t s .  A d i s c u s s i o n of these  phenomena i s beyond the scope o f t h i s t h e s i s and, f u r t h e r m o r e ,  13 i s i r r e l e v a n t s i n c e we of the gas.  The  are concerned o n l y w i t h the f i n a l s t a t e 2  d i s c u s s i o n t h e r e f o r e a p p l i e s only to gas  out-  s i d e the t r a n s i t i o n zone.  2.3  Shock Waves i n R e a l Monatomic Gases The  theory  o u t l i n e d above a p p l i e s to p e r f e c t gases.  However, the r e s u l t s p r e d i c t w e l l the behaviour of the monatomic gases up"to temperatures o f about 8000°K. t h i s p o i n t , i o n i z a t i o n and, be  considered  Equations f o r the  to a l e s s e r e x t e n t ,  i n d e t e r m i n i n g the p r o p e r t i e s  (2.1)  to  (2 .'3) s t i l l  i n t e r n a l energy and  apply.  inert  Beyond  e x c i t a t i o n must  of s t a t e  However, the  2. expressions  the equation of s t a t e must be  modified. We  are concerned w i t h temperatures at which only  s i n g l e stage o f i o n i z a t i o n need be c o n s i d e r e d . volume V of gas which has temperature.  We  e l e c t r o n s by N^,  been heated by  N, +  and N  g  = N  e  N = N introduce  E l e c t r o n s and  a  (2.17)  +  + N  A  i s defined  by (2.18)  +  =  e  W  N  =  N  fl  e I  N.  With these d e f i n i t i o n s , the equation of s t a t e f o r the ne  K  ions  the degree of i o n i z a t i o n , a, N  gas. i. s (7,9,10,11) * *  and  that  t o t a l number of heavy p a r t i c l e s , N,  now  a shock wave to such a  respectively.  N  We  Consider a  denote the numbers of n e u t r a l atoms, i o n s ,  must be produced i n p a i r s so  The  a  ( 2  '  1 9 )  ionized  14 pV = ( N  + N  A  + N )kT  +  e  = (1 + a)NkT  (2.20)  I t has been assumed t h a t the component  gases o f e l e c t r o n s ,  atoms and ions are i n mutual e q u i l i b r i u m at temperature T. The mass d e n s i t y , p, i s g i v e n by P  V  = N m A  A  + N m +  +  +N  e e m  ri Nm. A s i n c e m << m. e A  (2.21)  n r . Thus + p  .= (1 + a) pRT  (2.22)  s i n c e R = k/m^. The i n t e r n a l energy, U, per u n i t volume, must now  i n c l u d e the i o n i z a t i o n a l energy.  I f the i o n i z a t i o n p o t e n t i a l  o f the atom i s I ,  U=  (I N I<T+ f N+kT + | N kT + N l ) V"' A  e  +  or  Here the c o n t r i b u t i o n s  from e l e c t r o n i c e x c i t a t i o n of both n e u t r a l  atoms and ions has been n e g l e c t e d . way  The e r r o r i n t r o d u c e d  i s l e s s than 2% f o r temperatures l e s s than 16,000°K  Equations  (2.1),  (2.2),  (2.3),  i n this W  .  (2.22) and (2.23) now s p e c i f y the  s t a t e o f the shock heated gas i n terms o f a. between a and the temperature and d e n s i t y thermal e q u i l i b r i u m i s p r o v i d e d  The r e l a t i o n s h i p  f o r an i o n i z e d gas i n  by the Law of Mass A c t i o n o r ,  as i t i s f r e q u e n t l y c a l l e d when expressed i n the f o l l o w i n g form, the Saha  equation:  Net _ / - a  -  Z Z+  z e  A  - / k T I  €  (2.24)  Here, Z^,  Z  and Z  +  are the p a r t i t i o n f u n c t i o n s of the r e s p e c t i v e  g  s p e c i e s , and are found f o r argon Z  e  = 2 K  to be  e  Z. = K. A A f  .....  Z  where K  and  The  +  .  = (4  2 e-  2 0 6  ^ ^'k T J  =  V = h = k '= nu =  +  °/ )K T  4  +  v  volume occupied by the gas Planck's constant Boltzmann's constant p a r t i c l e mass of s p e c i e s i ( i  = A,+ ,e)  c o n t r i b u t i o n t o Z^ from e l e c t r o n i c e x c i t a t i o n has been  n e g l e c t e d , w i t h an e r r o r of l e s s than 11 f o r temperatures 1 5 , 0 0 0 K ^ * ** ^ . o  2  The  expression for Z  from the l o w - l y i n g f i r s t v a l i d up to 2 0 , 0 0 0 K o  *  ( 1 0  1 1 , 1 2 : )  ,  .  Combining  ^ . M  to  includes contributions  +  e x c i t e d s t a t e o f the argon '  up  e  -  i o n and i s  (2.20) and I  /  k  T  j '  /  (2.24) 2  C2.2S)  When i o n i z a t i o n i s important, then, i n a d d i t i o n to the Rankine-Hugoniot e q u a t i o n s , (2.1) m o d i f i e d equation of s t a t e ,  to (2.3), we must use  (2.22), and i n t e r n a l energy,  t o g e t h e r w i t h Saha's e q u a t i o n f o r a, (2.24). complete s e t of equations equations.  (2.23),  shall call  this  the Augmented Rankine-Hugoniot  Knowing s t a t e 1 and the shock speed, they u n i q u e l y  specify state  2.4  We  the  2,  Dependence of Shock S t r e n g t h on Diaphragm Pressure R a t i o The  degree to which the t e s t gas  i s shock heated depends  on the shock s t r e n g t h which, i n a c o n v e n t i o n a l shock  tube,  16 depends on the desirable  diaphragm p r e s s u r e r a t i o p^/p^.  to know the  Mach number, M,  d r i v e r gas  r e l a t i o n s h i p between P^/p^  or the  t h i s dependence we when the  diaphragm b r e a k s .  test section.  p r e s s u r e i n the  To  We  flow.  Due  Driver  to the  the  s h a l l assume t h a t  the  therefore gas  depletion  d r i v e r s e c t i o n near the  a r a r e f a c t i o n wave, or f a n .  propagates i n t o the some time, t ^ , the Quantities we  i n the  denote by  across the  the  d r i v e r gas  introduces  then s t a r t s to move of d r i v e r gas,  diaphragm drops.  head of the  expanded d r i v e r gas subscript  interface  3.  = p  sound speed, a^.  behind the  and  3 through the  In t h i s case i t can be • through the  v +  After  2-2.  contact  driver  (2.26)  3  p^.  For  a perfect the  we  then  shown  require  d r i v e r gas  of  t r a n s i t i o n from s t a t e  r a r e f a c t i o n wave i s an i s e n t r o p i c  \-\  surface  c o n t a c t s u r f a c e i s continuous:  c o n s t a n t r a t i o of s p e c i f i c h e a t s , to s t a t e  gas  Since there i s no momentum f l u x  f i n d the dependence of shock s t r e n g t h on p^, r e l a t i o n between p^  This  driver  (assumed p l a n a r ) between t e s t and  2  the  r a r e f a c t i o n wave  s i t u a t i o n i s as shown i n F i g u r e  p  the  The  at the  gases, the p r e s s u r e across the  To  determine in  decrease i n pressure propagates backwards through the as  shock  must c o n s i d e r the p r o c e s s e s o c c u r r i n g  p e r t u r b a t i o n s i n t o the  i n t o the  and. the  shock p r e s s u r e r a t i o , p2/p^.  diaphragm disappears i n s t a n t a n e o u s l y and no  It i s therefore  process.  that =  constant  r a r e f a c t i o n w a v e ^ , where v i s the  4  (2.27) flow speed  and  17 a the sound speed.  Since i n i t i a l l y 4  V  and s i n c e  v  =  °»  (2.28)  2 ~ 3  (2.29)  V  f o r the same reason t h a t p  2  = Pg, we have t h a t z Of  For an i s e n t r o p i c  I  process f>j> =  so t h a t , u s i n g  -  constant  (2.31)  (2.13),  2&V •fey  _  / a * \* v  We then o b t a i n the r e l a t i o n between p^ and p  (2.32)  2  Zh  A The  - /  \  (2.33)  d e s i r e d dependence o f shock s t r e n g t h on P^/p^  solving  (2.33)  equations  i s obtained by  together w i t h the Augmented Rankine-Hugoniot  of s e c t i o n 2.2.  For a p e r f e c t gas we f i n d the  implicit  relationship  nil  -  (2 34)  T h i s e x p r e s s i o n shows t h a t there i s a l i m i t t o the value o f M which can be produced f o r given d r i v e r and t e s t gases. p e r f e c t gas, as V^/v± approaches  infinity,  For a  18  i  4 a  4  ^ ~  a  >  3" 3<3— V  R a r e f a c t i o n Diaphragm Wave Position Head Tail (a) Wave System i n Shock Tube  1 p  1  1  V2  Contact Surface  — >  V  s  Shock Front , .  i 1  P  1 1  1  2  3  1 1 1  3  .1  s  I  1 1 1 1 1  *>2 • I  Pi  1  (b) P r e s s u r e D i s t r i b u t i o n i n Shock Tube  (c) x - t Diagram  0  F i g u r e 2-2 Gas Flow i n Shock Tube A f t e r Diaphragm B u r s t s  x  o r , to a good approximation,  ri-tW To o b t a i n strong shocks one sound speed, a^, at h i g h  2.5  S o l u t i o n of the The  forward  ^'f;  C2.35)  thus r e q u i r e s a d r i v e r gas  of high  pressures.  Equations  s o l u t i o n of the equations  f o r a p e r f e c t gas.  i s t e d i o u s but s t r a i g h t -  For a monatomic gas  including ion-  i z a t i o n e f f e c t s the s o l u t i o n becomes much more d i f f i c u l t . have used a d i g i t a l computer to s o l v e the equations using the i t e r a t i o n technique The  We  numerically-  d e s c r i b e d by Gaydon and H u r l e ^ .  c a l c u l a t e d dependences o f P2/p^,  P2/Pi> ^2^1'  *  anc  a  o  n  M  '  and the dependence of M on p^/p^, are shown on the f o l l o w i n g pag for  argon t e s t gas  and  These curves  f o r a d r i v e r gas  enable the p r o p e r t i e s of the gas  to be p r e d i c t e d from the i n i t i a l number.  The numbers beside  p r e s s u r e , p^, The  of constant  s t a t e 1 and  -  1.67.  i n state 2  the shock Mach  each curve s p e c i f y the i n i t i a l  gas  i n Torr.  lowest  curve i n Figure 2-7  (a^/a^ = 3.3)  shocks i n room-temperature argon d r i v e n by helium.  %^  In t h i s case i t i s impossible  i n excess of about Mach 12.6.  p e r t a i n s to  room-temperature  t o achieve  shock speeds  By h e a t i n g the d r i v e r gas,  however, i t s sound speed can be r a i s e d and f a s t e r shocks can obtained  as shown by the other curves  of Figure  2-7.  be  20  1000  1 F i g u r e 2-3  5  10 15 Mach Number  20  P r e s s u r e R a t i o a c r o s s Shocks i n Argon  25  21  14  J  Mach Number F i g u r e 2-4  D e n s i t y and V e l o c i t y R a t i o s A c r o s s  Shocks i n Argon  22  20  H  = 1 0 0 Torr 18  T  R  =  293"K  16  14  12  10  co  o  t-i  CM  8  H  4  H  -j  1  1—i—t-—r—i—s—r 5  I  10  I  >  I  Mach Number  >  I  t  <  15  F i g u r e 2 - 5 Temperature B e h i n d Shock i n Argon  t  I  20  i  I  Shock Mach Number F i g u r e 2-6  Degree o f I o n i z a t i o n  (a)  i n Shock-Heated Argon  log F i g u r e 2-7  1 0  (p /p ) 4  1  Shock Mach Number (M) as a F u n c t i o n o f Diaphragm R a t i o (P4/Pi) i n Argon D r i v e n by H e l i u m  f o r Shocks  2.6  The Shock Heated  Gas  Our assumption t h a t the gas of r e g i o n thermodynamic  2 i s i n a s t a t e of  e q u i l i b r i u m deserves f u r t h e r comment.  gas possesses only  t r a n s l a t i o n a l degrees of freedom.  A perfect These are  e x c i t e d i n the t r a n s i t i o n zone we have i d e n t i f i e d w i t h the shock f r o n t i n s e c t i o n 2.1.  Since energy i s exchanged v e r y e f f i c i e n t l y  between p a r t i c l e s of the same mass, a s t a t e of t r a n s l a t i o n a l equilibrium i s established within can now  a v e r y few c o l l i s i o n s .  We  d e f i n e more p r e c i s e l y the shock t r a n s i t i o n as the  over which t h i s t r a n s l a t i o n a l e q u i l i b r i u m  i s achieved.  Real gases, however, possess i n t e r n a l as w e l l as l a t i o n a l degrees of freedom.  distance  trans-  In the t r a n s i t i o n zone a s t a t e o f  t r a n s l a t i o n a l e q u i l i b r i u m w i l l s t i l l be achieved very  q u i c k l y  I f the temperature c h a r a c t e r i z i n g t h i s s t a t e i s s u f f i c i e n t l y  1  0  low,  no o t h e r degrees o f freedom w i l l be e x c i t e d to a s i g n i f i c a n t extent and the s t a t e o f the gas w i l l be one of equilibrium.  However, i f t h i s temperature exceeds about  i n argon, f o r example, ibrium  thermodynamic  then f o r a s t a t e of thermodynamic  7000°K equil-  to be s e t up, some o f the t r a n s l a t i o n a l energy must be  d i s t r i b u t e d among the o t h e r a v a i l a b l e degrees o f freedom.  The  c h a r a c t e r i s t i c time f o r t h i s process i s c a l l e d the r e l a x a t i o n time and the d i s t a n c e  over which i t o c c u r s , the r e l a x a t i o n  A r e l a x a t i o n time may  be d e f i n e d  f o r each such p r o c e s s .  d i a t o m i c gases, f o r example, we may  zone.  For  speak o f r e l a x a t i o n times  for rotation, vibration, dissociation, electronic excitation, and i o n i z a t i o n .  A true temperature f o r the gas cannot be  u n t i l a l l the a p p r o p r i a t e  r e l a x a t i o n processes have been  defined completed'  and a s t a t e o f complete  thermodynamic e q u i l i b r i u m  thus  established. For argon we need c o n s i d e r  only  ionizational relaxation,  s i n c e the e x c i t a t i o n a l r e l a x a t i o n o f argon atoms and ions i s  (9 10 12") n e g l i g i b l e f o r temperatures  of interest here  v  '  '  J  .  By the  time the gas has passed through the shock t r a n s i t i o n a t r a n s lational equilibrium exists.  Through' the r e l a t i v e l y  inefficient  f13 - 191 process o f atom-atom c o l l i s i o n s ^ p a i r s are then produced. are  , a few e l e c t r o n - i o n  When a s u f f i c i e n t number of e l e c t r o n s  available, equilibrium  i o n i z a t i o n i s q u i c k l y brought  about  through e l e c t r o n - a t o m c o l l i s i o n s .  The extent of the r e l a x a t i o n  zone depends on the c r o s s - s e c t i o n s  f o r these p r o c e s s e s .  Some  d i s t a n c e behind the shock f r o n t , then, a s t a t e of thermodynamic e q u i l i b r i u m i s reached f o r which the Augmented equations apply. small  Rankine-Hugoniot  P r o v i d e d the width of the r e l a x a t i o n zone i s  compared to the shock f r o n t - c o n t a c t  useable s l u g o f e q u i l i b r i u m gas s t i l l  surface  exists.  separation, a  In t h i s  respect,  the presence of even small q u a n t i t i e s of e a s i l y i o n i z a b l e impurities  isbeneficial.  These  impurities  can r e a d i l y supply  e l e c t r o n s which then i o n i z e the argon atoms, so t h a t  equilibrium  i s brought about more q u i c k l y and the r e l a x a t i o n zone i s shortened. A snapshot o f a s t r o n g depicted is  i n Figure  followed  luminosity.  shock i n argon might  2-8. A sharp luminous  appear as  l i n e a t the shock f r o n t  by a dark r e l a x a t i o n zone and then a r e g i o n The sharp luminous  v i b r a t i o n a l spectra  of g e n e r a l  l i n e has been found to be due t o  o f such e a s i l y d i s s o c i a t e d molecules as C  ?  27  P ' 2  S> ' 2 T  2  Uniform Luminosity F i g u r e 2-8  ^  P  Relaxation Zone (Dark)  l '  Shock F r o n t ( F a i n t l y Luminous)  L u m i n o s i t y S t r u c t u r e i n S t r o n g Shock  28 and  CN.^^  these s p e c i e s r a d i a t e before  the h i g h temperatures.  they are destroyed by  In the dark r e g i o n the d i s s o c i a t e d  atoms a r e i o n i z e d and the e l e c t r o n s produced a c q u i r e energy through c o l l i s i o n s t o i o n i z e the argon atoms. of f a i r l y uniform  sufficient The r e g i o n  l u m i n o s i t y sets i n when the gas has reached  a s t a t e o f e q u i l i b r i u m , and emits a spectrum c h a r a c t e r i s t i c o f this state.  In extremely pure gases, the shock f r o n t l u m i n o s i t y  may be absent and the e q u i l i b r i u m r a d i a t i o n w i l l delayed.  2.7  ~  '  be c o n s i d e r a b l y  .  Boundary Layers In d e r i v i n g the shock r e l a t i o n s i n s e c t i o n s 2.1 and 2.2,  the e f f e c t o f the v e s s e l c o n t a i n i n g the gas was i g n o r e d .  Due to  v i s c o s i t y , gas at the shock tube w a l l must have zero v e l o c i t y i n the l a b o r a t o r y frame of r e f e r e n c e .  S i m i l a r l y , the temperature  of t h i s gas must be t h a t o f the w a l l . v e l o c i t y and temperature g r a d i e n t s  Consequently, r a d i a l  are e s t a b l i s h e d i n the shock  heated gas. The r e g i o n near the w a l l where these g r a d i e n t s are significant thickness  i s c a l l e d the boundary l a y e r .  The boundary l a y e r  i s f o r m a l l y d e f i n e d as the d i s t a n c e  which the r e l e v a n t q u a n t i t y  achieves  from the w a l l a t  99% o f i t s v a l u e  f a r from  the w a l l . Shock tube boundary l a y e r s are u s u a l l y f a i r l y pared t o the tube r a d i u s f o r pressures  t h i n com-  g r e a t e r than a few T o r r .  However, t h e i r e f f e c t can be s i g n i f i c a n t .  Some of the shock-  heated gas which i d e a l l y would be c o l l e c t e d between shock f r o n t and  contact  s u r f a c e remains i n the boundary l a y e r and i s passed  29 over by the c o n t a c t s u r f a c e .  Due  to the thermal boundary  layer,  the gas d e n s i t y i n steady s t a t e must be h i g h e s t near the w a l l s i n c e the r a d i a l p r e s s u r e boundary gas.  gradient i s small.  Therefore  the  l a y e r can c o n t a i n a l a r g e p o r t i o n of the shock heated  T h i s removal of gas reduces the t h i c k n e s s o f the shock-  heated gas s l u g between the shock f r o n t and c o n t a c t s u r f a c e . boundary  layer also introduces  v e l o c i t y i n t o the shock-heated The boundary  a s m a l l r a d i a l component of gas.  l a y e r a l s o has the e f f e c t of i n c r e a s i n g the  speed o f the c o n t a c t s u r f a c e above the p r e d i c t e d value a t t e n u a t i n g the shock f r o n t .  and  As shock-heated gas i s removed,  d r i v e r gas, and hence the contact s u r f a c e , must advance the shock f r o n t .  Therefore  the speed of the c o n t a c t  i n c r e a s e s above the i d e a l v a l u e . t h i s way,  p , must then drop as w e l l . 2  r e q u i r e s t h a t the shock speed decrease (see F i g u r e  shock a t t e n u a t i o n , s u c c e s s i v e a x i a l samples  and M thus  2-4).  Due  to the  of gas are heated to  For the low a t t e n u a t i o n u s u a l l y present  r e s u l t i n g a x i a l non-uniformity  The E f f e c t  2  pressure,  l a y e r s on the p r o p e r t i e s o f the  r e t a i n e d shock-heated gas are i n g e n e r a l s m a l l .  2.8  surface  The shock-heated gas  The r e l a t i o n between p  The e f f e c t of boundary  towards  As the d r i v e r gas expands i n  i t s p r e s s u r e , p^, drops.  l e s s e r degrees.  The  the  i n the s l u g i s n e g l i g i b l e .  of Shock Tube E x t r e m i t i e s  Up to t h i s p o i n t we have assumed t h a t the t e s t and d r i v e r s e c t i o n s are of i n f i n i t e  length.  We now  c o n s i d e r the r e f l e c t i o n  o f the shock and r a r e f a c t i o n waves from r i g i d w a l l s , f o r example,  30  .  the end w a l l s o f the d r i v e r and t e s t s e c t i o n s .  On r e f l e c t i o n  from a r i g i d w a l l , the wave nature i s n o t changed;  that i s , a  shock i s r e f l e c t e d as a shock and a r a r e f a c t i o n as a r a r e f a c t i o n . When the r a r e f a c t i o n r e v e r s e s i t s d i r e c t i o n o f motion on r e f l e c t i o n from the d r i v e r end w a l l , i t proceeds i n t o gas which has been s e t i n motion by the i n c i d e n t r a r e f a c t i o n .  The r e f l e c t e d  wave then propagates down the shock tube w i t h a speed g i v e n by the sum of the l o c a l flow v e l o c i t y and the l o c a l speed o f sound. I t must t h e r e f o r e e v e n t u a l l y overtake the c o n t a c t s u r f a c e and the shock f r o n t i f the t e s t s e c t i o n i s s u f f i c i e n t l y l o n g .  When i t  does s o , the p r e s s u r e , p , p r o p e l l i n g the shock wave decreases 2  and the shock wave a t t e n u a t e s .  T h i s p o i n t i s taken up again  i n Chapters 5 and 6. When a shock i s r e f l e c t e d  from a r i g i d w a l l , i t r e t u r n s as  a shock and t h e r e f o r e heats even more the gas which was p r o c e s s e d by the i n c i d e n t shock.  As a r e s u l t , the doubly-shocked gas may  become h i g h l y luminous even though the i n c i d e n t shock f r o n t i s too weak to be luminous. i d e n t i f y i n g non-luminous obstruction.  T h i s f a c t suggests a technique f o r shock f r o n t s by r e f l e c t i n g  them from an  The s t a t e o f the doubly-shocked gas may be c a l -  c u l a t e d from the s t a t e 2 u s i n g the theory o f s e c t i o n 2.2. As the r e f l e c t e d shock continues t o move back along the shock tube, i t meets the c o n t a c t s u r f a c e where, i n g e n e r a l , i t i s refracted.  The change i n speed o f the r e f l e c t e d shock wave on  p a s s i n g through the c o n t a c t s u r f a c e p r o v i d e s a convenient means of d e t e c t i n g the c o n t a c t s u r f a c e .  T h i s r e f l e c t e d shock technique  was a p p l i e d t o o b t a i n some r e s u l t s p r e s e n t e d i n Chapter 6.  31 CHAPTER 3  3.1  APPARATUS  Introduction  Figure  The  construction  of  3-1,  i s discussed  in this  operation  has  described  in section  cation. of  .  The  2 inches  assembled  been g i v e n  as  investigation  3.2  The  and  consists  s h o c k t u b e , shown i n  chapter.  of  The  i n sections  from s i x - i n c h square s l a b s  of  lucite  The  laminae supporting  electrodes  (2,  3,  this is  the  remaining  Figure  are  3.6  and  A-A') 3-4  fabridiameter  be  used  in  the  3.7.  shock tube as  small allows  and  gap  acetic  acid.  The  upper e l e c t r o d e  i n the  lucite  II)  and  back p l a t e  whole  can  be  assembly  taken apart  hole  through  evacuation  the  and the  d r i v e r s e c t i o n was  of  lower  milled  (1).  the  lead  O-rings  is bolted to  in  current  between  replace  after several  assembled filled  rod  The  driver sidewall  filling  3-2.  adjusting  section.  The  laminated  shown i n F i g u r e  i n t o a recess  shows a p h o t o g r a p h o f  p r e s e n t work, t h e  is  glued  joints.  (I and  A  inside  w h i c h may  i n s u l a t i n g sheet which d e t e r i o r a t e s  discharges. Section  lucite  Smy  together with  lead  i n t o a recess  brass plates Teflon  the  bonded  current  composite  glued  seal  are  and  an  is  Section the  electrode  for i t s  instrumentation  of  4)  has  of i t s  driver section  given  several pieces basic  description  The  d e t a i l s are  driver section  and  A  1.2.  ( s e c t i o n 3.5)  i s discussed  Driver The  3.2  desired.  Smy  in section  test section and  the  the hundred  (see  driver.  driver.  with helium  In at  the  Driver Section Backstrap  Test Gas  Section  Dump Chamber Pressure Gauges  Inlet /Flanged Copper P i p e  /Lk. if ;  Diaphragm  0-ring  Glass  Pipe  Vacuum Pump C a p a c i t o r Bank (50 j i f a r a d ) R i n g i n g P e r i o d — '21 p.sec 4  -1  L o g a r i t h m i c Decrement * 1.7*10 s e c  F i g u r e 3-1  Schematic Diagram o f V e r t i c a l S e c t i o n through  Shock Tube A x i s  F i g u r e 3-2  D r i v e r S e c t i o n o f Shock Tube ( S c a l e : Two-Thirds F u l l  Size)  34 atmospheric  pressure.  A i r was  used i n Smy's. o r i g i n a l  but the use of helium r e s u l t e d i n f a s t e r shocks. has  design  Since  helium  a low breakdotm v o l t a g e , a t r i g g e r a b l e a i r spark gap  was  r e q u i r e d i n s e r i e s w i t h the d r i v e r spark gap, which, i n our experiments,  3.3  The  had  a s e p a r a t i o n of 2  Discharge The  Circuit  discharge c i r c u i t  F i g u r e 3-3.  The  10-microfarad  cm.  i s shown s c h e m a t i c a l l y i n  c a p a c i t o r bank, C, c o n s i s t s of f i v e NRG  c a p a c i t o r s i n p a r a l l e l , and i s charged by  Type  203  a  Sorenson Model 1020-30 power s u p p l y , P, to a v o l t a g e o f 10 ± 0.1  k i l o v o l t s unless otherwise  stated.  A trigger  spark  a c t i v a t e d by the p u l s e generator, T, and t r a n s f o r m e r , X, produces a s m a l l degree of i o n i z a t i o n i n the a i r spark gap, A, causing i t to conduct. pleting  The  d r i v e r spark gap,  the c i r c u i t .  D, then breaks  down, com-  T represents a thyratron pulse  generator  and v o l t a g e doubling c i r c u i t which have been d e s c r i b e d (22 elsewhere  231 *•  v  J  .  p u l s e of amplitude 3.4  The  I t may  be t r i g g e r e d manually or by a p o s i t i v e  g r e a t e r than 20 v o l t s .  Diaphragm The  diaphragm i s s e c u r e l y clamped between the d r i v e r  a f l a n g e d two-inch  copper p i p e which forms the f i r s t  the shock tube ( F i g u r e 3-1).  s e c t i o n of  0-rings on each s i d e of the  diaphragm g r i p i t and p r o v i d e a vacuum-tight s e a l .  After  discharge  the clamp i s lossened i n order to r e p l a c e the diaphragm and retightened.  and  then  D  A: C: D: P: T: X: L:  A i r Spark S w i t c h C a p a c i t o r (.50 j i f a r a d ) Shock. Tube D r i v e r S e c t i o n 20 K i l o v o l t Power Supply T r i g g e r Pulse Generator I s o l a t i o n and Step-Up T r a n s f o r m e r C u r r e n t Leads (0.063 i n c h t h i c k copper s h e e t 4 i n c h e s wide and 3 f e e t l o n g )  F i g u r e 3-3  D e t a i l s of Discharge C i r c u i t  36 In Smy's o r i g i n a l shock tube, mylar diaphragms about 0.0015 i n c h t h i c k were used. diaphragm b u r s t and tube. deposit  Upon f i r i n g  the d i s c h a r g e ,  tore i n t o s m a l l p i e c e s which flew down the  These p i e c e s burned i n f l i g h t on the shock tube w a l l .  and  l e f t a dense b l a c k  In our experiments the  b i t s o f mylar a l s o damaged d e l i c a t e pressure used as measuring d e v i c e s .  fragments a f t e r only three  s e v e r e l y chipped  or four s h o t s .  burning  probes which were  A h a l f - i n c h t h i c k g l a s s end  used i n some experiments was  damage and  the  To  by  plate  the mylar  avoid t h i s k i n d  to improve the c l e a n l i n e s s o f the shock tube, the  mylar diaphragms were r e p l a c e d by t h i n sheets  of shim b r a s s .  These remained i n one p i e c e a f t e r r u p t u r i n g and  l e f t only  a  s l i g h t d e p o s i t which c o u l d e a s i l y be removed w i t h t i s s u e . mylar d e p o s i t c o u l d not be completely The  brass  inches  removed without u s i n g  a sharp k n i f e along  o f each d i a g o n a l  than the diaphragm t h i c k n e s s . by v a r y i n g the f o r c e exerted effect  The  depth of the scores  The  the  no  - varied  noticeable  along both d i a g o n a l  A photograph of diaphragms before 3-5.  and  cuts.  after bursting is  f o u r t r i a n g u l a r shaped s e c t i o n s of  b u r s t diaphragm are c a l l e d " p e t a l s " . were clamped a g a i n s t the open end  Initially  of the  the  opening was  f l a n g e d copper pipe  f i t t e d with  the  diaphragms  diaphragms t h i n n e r than 0.008 i n c h , the p e t a l s t o r e o f f .  prevent t h i s , the pipe  of  the grooves were deep enough  t h a t the diaphragms opened completely  for  acid.  to a depth s l i g h t l y l e s s  on the k n i f e - had  on the shock speed p r o v i d e d  shown i n Figure  The  diaphragms c o n s i s t e d of t h r e e - i n c h squares  automotive shim stock s c r i b e d w i t h c e n t r a l two  of  a brass  insert  but To  F i g u r e 3-4  Two Views of The D r i v e r S e c t i o n  F i g u r e 3-5  Diaphragms B e f o r e and A f t e r B u r s t i n g  38 w h i c h had to  a 1.4  merge w i t h  downstream. square  i n c h square  the The  edges and  c i r c u l a r pipe  fitting to  the  remained  flanged  when p r o p e r l y  scribed.  t h i c k d i a p h r a g m s were  Test  from the  a b o u t two  meters  test  diaphragm. long  of copper p i p e , pyrex p i p e ,  test  joints  this . inch  pipe  of  The  was  ranging  in  A l l opened  by  attached  thickness  satisfactorily  s t a t e d , 0.005  inch  i s admitted  first  hole  s p a n n e d by  s l a b s bonded electrically  inside  tube  and  machined  lucite  0-rings.  The  mentioned  through a small hole Seven r a d i a l  through the w a l l  these  inches  holes  together grounded  and  from  t o the  first  which lengths clamp-  part  of  the  previously. i n the  side  spaced  to admit the  from  are  tube  sections,  holes  i s enclosed  i n order  a  diameter through  i s constructed  copper p i p e  is five  distances  s e c t i o n c o n s i s t s of  The  v i a a needle valve.  length  lucite  pipe  gas  This  sealed with  i n t e r v a l s were b o r e d  probes. pipe  and  section is a flanged  Argon t e s t  aperture  section, a l l axial  of two-inch  s h o c k wave p r o p a g a t e s .  at the  kilovolts.  Section  measured  ed  o f 13  thinner  shock tube a x i s  locating  otherwise  the  used.  I n d i s c u s s i n g the  the  the  Diaphragms  Unless  inches  f o r diaphragms  i n excess  on  were t r i e d .  two  tapered  f o l d e d back along  voltages  a t h r e e - i n c h square  copper p i p e .  d i a p h r a g m and  at a d i s t a n c e  except  d i a p h r a g m s were c e n t e r e d  them i n t o  The  walls  attached  discharge  f r o m 0.002 i n c h t o 0.015  3.5  a t the  diaphragm p e t a l s then  t h a n 0.003 i n c h and The  opening  at  of two  pressure  diaphragm.  The  i n a c a s i n g made  copper p i p e .  to s h o r t - c i r c u i t  The any  39 c u r r e n t s which might pass  from the dr:.ver e l e c t r o d e s through the  i n t e r v e n i n g conducting gas. A s p e c i a l l y c o n s t r u c t e d p o r t i o n o f the shock tube i s shown i n F i g u r e 3-6.  P o r t h o l e s , which c o u l d be plugged when  not i n use, allowed access t o the shock tube i n t e r i o r . f i r s t p o r t was u s u a l l y occupied by a p r e s s u r e probe.  The On the  o p p o s i t e s i d e o f t h i s s e c t i o n , a s l i d i n g mount allowed the a x i a l s e p a r a t i o n between a p r e s s u r e probe and the second p o r t t o be v a r i e d c o n t i n u o u s l y over a t h r e e - i n c h e x t e n t . S e v e r a l l u c i t e d i s c s 1.5 i n c h t h i c k and 6 inches i n diameter w i t h two-inch  a x i a l holes could be i n s e r t e d as d e s i r e d  between g l a s s s e c t i o n s of the shock tube.  A r a d i a l h o l e i n each  d i s c allowed a p r e s s u r e probe t o be i n s e r t e d . The  shock tube was terminated i n a s o - c a l l e d dump  chamber (see F i g u r e 3-1) c o n s t r u c t e d from a f o u r - i n c h l e n g t h o f f o u r - i n c h diameter  brass pipe".  A removable end p l a t e allowed the  shock tube t o be cleaned w i t h a long brush. to  Separate  connections  the vacuum pump and p r e s s u r e gauges were made through the  s i d e - w a l l o f the dump chamber. The mechanical and  shock tube was evacuated w i t h a Cenco Hyvac 14 pump.  The lowest pressure achieved was 0.5 m i l l i t o r r  the leak r a t e was about 20 m i l l i t o r r p e r hour.  Pressures  were measured w i t h two Edwards Vacustat gauges c o v e r i n g the ranges  0 t o 1 T o r r and 0 t o 10 T o r r and w i t h two Edwards capsule  d i a l gauges c o v e r i n g the ranges  0 t o 40 T o r r and 0 to 760 T o r r .  A f t e r e v a c u a t i o n the tube was f l u s h e d w i t h argon b e f o r e to  the d e s i r e d p r e s s u r e .  filling  /Neoprene  Gaskets Plug  Plug Lucite  3L Port  Pyrex Pipe  Port  1  2  Diameter Slot  r—  0.25"  :  wide,  3"  long  i O  Lucite  T  Figure  4.25  3-6  inch  square  Horizontal Axial  Brass''  Section  through  Special Test  S l i d i n g Mount f o r Pressure Probe  Section  41 3.6  Instrumentation The shock f r o n t was d e t e c t e d w i t h p i e z o e l e c t r i c p r e s s u r e  probes.  The d e s i g n c r i t e r i a and c o n s t r u c t i o n of the probes are  d i s c u s s e d i n the Appendix. the  A p p l i c a t i o n o f an a x i a l s t r e s s t o  a c t i v e element o f the probe produces an e l e c t r i c a l  which i s r e c o r d e d on an o s c i l l o s c o p e . is  shown i n F i g u r e 3-7.  signal  A g e n e r a l purpose probe  A s i m i l a r t r i g g e r probe was u s u a l l y  mounted i n the f i r s t p o r t o f the shock tube s e c t i o n of F i g u r e 3-6. Shock speeds c o u l d be a c c u r a t e l y measured i n the f o l l o w i n g way.  The s i g n a l from the t r i g g e r probe was a m p l i f i e d by one  channel of a Tektronix-Type 1A1 P l u g - i n U n i t and then used to t r i g g e r timebase B o f a Type 545A o s c i l l o s c o p e .  The d e l a y e d  t r i g g e r p u l s e was used t o t r i g g e r timebase A which d i s p l a y e d the s i g n a l from a second p r e s s u r e probe some d i s t a n c e downstream o f the  first.  T h i s t i m e - o f - f l i g h t measurement,, when c o r r e c t e d f o r  the  d e l a y s i n h e r e n t i n the probes  (see Appendix), was used to  c a l c u l a t e an average shock speed over the d i s t a n c e between the probes. A smear camera (designed and c o n s t r u c t e d by J . P. Huni) w i t h w r i t i n g speed v a r i a b l e between 45 ysec/cm  and 4 ysec/cm was  used-to f o l l o w the p r o g r e s s o f luminous shock wave f e a t u r e s . The image was recorded on P o l a r o i d f i l m and p r e s e n t s an x - t diagram of the flow i n the shock tube.  The d r i v e r d i s c h a r g e was  t r i g g e r e d by a p u l s e generated when the smear camera r o t a t i n g m i r r o r had achieved a p r e s e l e c t e d speed and was i n the r e q u i r e d p o s i t i o n to r e c o r d the event o f i n t e r e s t . s l i t was imaged  The smear camera  on the f i l m p l a n e by a f r o n t - s i l v e r e d  concave  BNC Connector i  Soft Plastic . Tube  B r a s s Case  .  I  to  Figure 3-7  S i m p l i f i e d Diagram of P r e s s u r e Probe and Housing  m i r r o r which o b v i a t e d camera.  the need f o r a lens system i n s i d e the  A p a i r of 15 cm  f o c a l l e n g t h lenses served  to image  the event onto the smear camera s l i t .  3.7  E l e c t r i c a l Measurements To measure the d r i v e r v o l t a g e , a T e k t r o n i x P6013A 1000:1  p o t e n t i a l d i v i d e r was  connected across the e l e c t r o d e  d i r e c t l y below the d r i v e r s e c t i o n . p o t e n t i a l d i v i d e r was  used  from the  c u r r e n t , a Rogowski c o i l  was  211 '  v  output  d i s p l a y e d on a c a l i b r a t e d o s c i l l o s c o p e .  To measure the discharge (20  The  leads  J  .  T h i s device  c o n s i s t s of a t o r o i d a l l y wound m u l t i -  t u r n c o i l whose minor r a d i u s i s small compared to i t s major radius.  The  r a p i d l y - v a r y i n g c u r r e n t to be measured i s passed  through the torus opening.  A p p l y i n g Ampere's law to a c l o s e d  path t h r e a d i n g the turns of the c o i l of magnetic i n d u c t i o n to be t u r n , one  obtains  where  n A I yo t  = = -  o u t  "=  v o l t a g e o f the  number of turns per u n i t l e n g t h of area of the i n d i v i d u a l turns d i s c h a r g e c u r r e n t to be measured magnetic p e r m e a b i l i t y constant time  coil,  of i n t e r e s t ,  1, where R and C are the r e s i s t a n c e and  the i n t e g r a t i n g c i r c u i t one  then obtains  individual  coil  and  coil  i s i n t e g r a t e d to g i v e the c u r r e n t , I.  Providing that f o r a l l frequencies >>  variation  - n A A . j l  U s u a l l y the output  (OCR  assuming the  small over the area of an  f o r the output  V  and  , LcO  <<  capacitance  L i s the s e l f - i n d u c t a n c e of the  R  and of  44  V  f  Vout  The  --  n  TIC  T  (3.2)  Rogowski c o i l used i n t h i s work was c o n s t r u c t e d  a s u i t a b l e l e n g t h o f RG 65A/U delay cable w i t h the outer ductor  removed.  Before  t o r o i d a l l y deforming t h i s c o i l  from  con-  i t was  wrapped w i t h a 0.001 i n c h t h i c k p i e c e o f brass  shim s t o c k , which  did  p e n e t r a t i o n o f the  not c l o s e on i t s e l f e l e c t r i c a l l y to permit  magnetic f i e l d .  T h i s s h i e l d was e l e c t r i c a l l y grounded t o  reduce the e l e c t r o s t a t i c c o u p l i n g between the c o i l and c u r r e n t leads.  To reduce f u r t h e r the e f f e c t o f such c a p a c i t a t i v e  c o u p l i n g , the c o i l  s i g n a l was a m p l i f i e d d i f f e r e n t i a l l y .  c o i l used, the c u r r e n t measured i s g i v e n by  I -- 1.1$ -10* V  o u t  j%ff-  For the  45 CHAPTER 4  4.1  THE  DRIVER  Introduction In t h i s chapter the nature of the d r i v i n g f o r c e  r e s p o n s i b l e f o r the h i g h shock speeds o b t a i n e d i n the Smy tube i s i n v e s t i g a t e d . opening  A technique  i n which we  use  shock  the diaphragm  process to measure the p r e s s u r e o f the d r i v e r gas i s  then d i s c u s s e d .  4.2  The  D r i v i n g Mechanism Smy^'^  a t t r i b u t e d the h i g h shock speeds t o e l e c t r o -  magnetic f o r c e s produced by the " b a c k s t r a p " c o n f i g u r a t i o n . There remains, however, another p o s s i b l e d r i v i n g mechanism. Chapter  2 i t was  In  shown t h a t very f a s t shocks can be produced i n  p r e s s u r e - d r i v e n shock tubes by u s i n g a d r i v e r gas o f h i g h sound speed a t h i g h p r e s s u r e .  J u s t T t h i s s i t u a t i o n may  be produced by  the e l e c t r i c a l d i s c h a r g e which c l e a r l y must i n c r e a s e the temperat u r e , and hence the p r e s s u r e and sound speed o f the d r i v e r by J o u l e - h e a t i n g . two  To i n v e s t i g a t e the r e l a t i v e  gas,  importance of the  p o s s i b l e shock d r i v i n g mechanisms s e v e r a l experiments were  carried  out. The  e x i s t e n c e of a downstream-directed  d r i v i n g f o r c e o b v i o u s l y r e q u i r e s the presence  electromagnetic of the  i n the absence of any other source of magnetic f i e l d of the d i s c h a r g e .  "backstrap" at the  Hence the removal o f the " b a c k s t r a p "  site  should  remove the d r i v i n g f o r c e and so have a very n o t i c e a b l e e f f e c t on the shock speed.  The  " b a c k s t r a p " was  e l i m i n a t e d by  conducting  46 the c u r r e n t from the upper d r i v e r e l e c t r o d e through s t r i p down each s i d e of the d i s c h a r g e chamber.  a copper  In t h i s  config-  u r a t i o n there i s no downstream-directed e l e c t r o m a g n e t i c f o r c e and so the shocks must be d r i v e n by the k i n e t i c p r e s s u r e of the d r i v e r gas heated w i t h and without  by the d i s c h a r g e .  Measurements o f shock speed  the " b a c k s t r a p " are compared i n F i g u r e 4-1  an argon p r e s s u r e of 1 T o r r at x = 126  cm.  The  for  removal of the  " b a c k s t r a p " i s seen to have only a s m a l l e f f e c t on the shock speed. Another experiment was  conducted  magnetic f o r c e s are n e g l i g i b l e .  Since an  d r i v i n g f o r c e would depend on the square  to c o n f i r m that e l e c t r o electromagnetic of the d i s c h a r g e  c u r r e n t , the speed o f shocks d r i v e n i n t h i s way s t r o n g l y on the c u r r e n t . d i s c h a r g e c u r r e n t was the d i s c h a r g e c i r c u i t .  With the " b a c k s t r a p " i n p l a c e , the  reduced by i n s e r t i n g s e r i e s i n d u c t o r s i n t o Each i n d u c t o r c o n s i s t e d of a c i r c u l a r  c o i l of two-inch wide copper s t r i p supported diameter  plywood d i s c .  should depend  Inductors  of one  the peak d i s c h a r g e c u r r e n t by 21% and  by a s i x - i n c h  and two  turns  49% r e s p e c t i v e l y .  reduced To  check t h a t the f i e l d c o n f i g u r a t i o n around the i n d u c t o r s d i d not cause erroneous  c u r r e n t measurements, the Rogowski c o i l  moved f a r t h e r away from the i n d u c t i v e loop and r e l a t i v e to the plane of the loop was  altered.  was  i t s orientation No  change i n the  c u r r e n t waveform r e s u l t e d . The  r e s u l t s of these measurements, shown i n F i g u r e  4-2,  demonstrate t h a t the shock speed does not depend on the d i s c h a r g e current.  I t may  be concluded,  then, t h a t e l e c t r o m a g n e t i c f o r c e s  5  H  o CO  >  4  T e s t Gas  s  Pressure = 1 Torr  Error Bars Represent Standard Deviations f o r at least 3 Shots  QJ <D ft  co  O  ,M o o  x  CO  o 4 7  8 Capacitor  Figure  4-1  Shock  -I— ID  9  Speeds With  Bank  Voltage  and Without  —T—  — S —  11  12  Backstrap Sidestraps  (Kilovolts) the Backstrap  —r— 13  - i —  14  — r 15  49 play by  an unimportant  the e l e c t r i c a l  role  and t h a t  discharge  the heating  of the driver  i s responsible f o r the high  gas  shock  speeds. In  a l l subsequent measurements, the "backstrap"  u r a t i o n was u s e d .  4.3  raised  t o determine  convenient  means  The  isotropic plates insert. the  of measuring  cuts  this  i s assumed  hinged  p.  discharge,  of the driving  provides  i n splitting  i s neglected.  a  quantity.  to split  The p e t a l s  i t i s  pressure.  instantaneously  the action of a  a r e assumed  the diaphragm  The e q u a t i o n  of motion  along  constant  t o behave  a t t h e edges o f t h e square-mouth  The work done  petals  a r e d r i v e n by gas  opening process  and t o open under  pressure,  freely  the shocks  t h e magnitude  of the diaphragm  diaphragm  diagonal  Pressure  temperature by the e l e c t r i c a l  observation  both  o f t h e D r i v e r Gas  established that  to high  desirable The  ;  Determination Having  config-  like  shock  tube  and bending of a petal i s  then  I where  P I 6  t  is is the is is  the a p p l i e d torque t h e moment o f i n e r t i a o f t h e p e t a l a b o u t appropriate axis t h e a n g l e t h r o u g h w h i c h t h e p e t a l moves t i m e ( t = 0 a t e = 0)  (4.1)  F i g u r e 4-3  Diaphragm P e t a l Geometry  Since the t o t a l f o r c e a c t i n g on a p e t a l i s p.area the torque about the a x i s 00'  3 i s ph /3  = ph ,  ( F i g u r e 4-3), w h i l e the  moment of i n e r t i a of the p e t a l about t h i s a x i s i s mh 2  7  /6, where  m i s the p e t a l mass g i v e n by m = ph d f o r a diaphragm of mass d e n s i t y p and t h i c k n e s s d.  Equation  d0 a  dt  "  1  (4.1)  then becomes  ZP  -  (4.2)  fhd  I n t e g r a t i n g t h i s equation s u b j e c t t o the c o n d i t i o n s 6=0, yields  Now  the time, t  de/dt = 0, at t = 0  (4.3)  e=  (4.4)  , f o r a diaphragm to open completely i s  o b t a i n e d by s e t t i n g e = IT/2:  L  A measurement o f t  Z  P  \  p  (4.5)  J  then p r o v i d e s a v a l u e f o r p. op  The  o  opening  r  A  time, t p , was  set up of F i g u r e 4-4.  Q  The  measured w i t h the smear camera  camera s l i t was  focused on a l i n e  Driver  Diaphragm  Shock Tube  Glass  Plate  Front-Silvered Mirror  Lens /  (a) E x p e r i m e n t a l  Arrangement  Entrance S l i t of Smear Camera  Region of Diaphragm Observed w i t h Smear Camera  (b) Diaphragm F i g u r e 4-4 Arrangement f o r Observing  F i g u r e 4-5  Diaphragm Opening  Smear Photograph of Diaphragm Opening  passing  through the center of the diaphragm at 45°  the d i a g o n a l s .  L i g h t from the hot d r i v e r gas  opening diaphragm.  shows the dependence of t  voltage, V  , f o r diaphragms 0.005 i n c h t h i c k . W  i l l u m i n a t e d the  A t y p i c a l smear photo i s shown i n F i g u r e  F i g u r e 4-6  •  to each of  4-5.  on the c a p a c i t o r bank One  would expect  "v.  the d r i v e r p r e s s u r e  to be p r o p o r t i o n a l to the c a p a c i t o r bank 2  energy d i s s i p a t e d i n the d r i v e r gas bank energies  (V  c o l d d r i v e r gas and  -  10  The  thus i n agreement w i t h equation 1/2  and  r a d i a t i v e energy  p r o p o r t i o n a l i t y of t (4.5).  and d ' , f o r constant  F i g u r e 4-7  to V ^  and  4-7  „where c = 9.28  The p r o p o r t i o n a l i t y  bank v o l t a g e shown i n (4.5).  y i e l d the e m p i r i c a l r e s u l t t = cd /V op c -1/2  (4.6)  1 / 2  volt-sec-cm  ' .  is  c  a l s o agrees with the p r e d i c t i o n o f equation  F i g u r e s 4-6  low  i s comparable t o the e l e c t r i c a l energy added,  at h i g h e n e r g i e s , where conductive  between t  V ) , except at  v o l t s ) , where the thermal energy of the  l o s s e s become a p p r e c i a b l e .  4-7  (i.e. p  Moreover, s i n c e F i g u r e s 4-6  and  seem to v e r i f y the adequacy of the t h e o r y , a s i n g l e measure-  ment of t  f o r a diaphragm of known t h i c k n e s s serves  to  q u a n t i t y , p, however, may  pressure  determine p. The  i n the d r i v e r , p^.  not be  the true  Because of complex flow p a t t e r n s  the d r i v e r as the diaphragm opens, the p r e s s u r e opening the p e t a l s , p, may  i n f a c t be  the s t a t i c d r i v e r p r e s s u r e , p^, bursts.  set up i n  effective in  c o n s i d e r a b l y l e s s than  immediately b e f o r e  the diaphragm  Furthermore, s i n c e the e x i t v e l o c i t y of d r i v e r gas  53  Y  0.05  0.10  V" F i g u r e 4-6  Dependence  1  0.15  ((Kilovolts) ) - 1  o f Diaphragm Opening Time on V o l t a g e  depends on the d r i v e r sound speed  (see Chapter 2, E q u a t i o n (2.30)),  one might expect the r e l a t i o n s h i p between p and p^ t o depend on this quantity  also.  To determine the r e l a t i o n between p and p^, the e x p e r i ment shown i n F i g u r e 4-8 was  set up.  The d r i v e r was  and then p r e s s u r i z e d s l o w l y u n t i l the diaphragm p r e s s u r e p^.  evacuated  b u r s t at  At t h a t time, l a s e r l i g h t passed through the  c e n t e r o f the diaphragm,  activating  a photodiode  which  t r i g g e r e d a T e k t r o n i x 551 d u a l beam o s c i l l o s c o p e .  L i g h t from an  i n t e n s e incandescent source passed through the diaphragm  opening  and was  shock  tube.  monitored by a p h o t o m u l t i p l i e r at the end of the  The p h o t o m u l t i p l i e r s i g n a l reached a maximum value when  the diaphragm  was  fully  open.  A pressure transducer ( A t l a n t i c  Research C o r p o r a t i o n Model LD-15) mounted e x t e r n a l l y a g a i n s t the shock tube w a l l produced the diaphragm open.  petals  a s i g n a l when the w a l l was  - that i s , when the diaphragm  A simultaneous r e c o r d of the p h o t o m u l t i p l i e r  transducer signals  i s shown i n F i g u r e 4-9.  o b s e r v i n g the diaphragm  op  d  -1/2 '  = Ap. ^4  fully  and  opening time are i n agreement. show that  -1/2 ' w i t h A determined by the method of l e a s t ' 1/2-1  squares to be 5.36 finds  was  The two methods of  R e s u l t s of many measurements (Figure 4-10) t  s t r u c k by  g ' cm  .  U s i n g e q u a t i o n (4.5), one  then  that p  4  = 2.55p  (4.7)  That i s , the a c t u a l d r i v e r p r e s s u r e i s 2.55 the p r e s s u r e e f f e c t i v e i n opening the  times l a r g e r  diaphragm.  than  56  Light Pipe  Photodiode  To Scope Trigger  Driver I  ,Mirror  Pressure Gauge Gas  Movie Sun-Gun Pressure Transducer  Scope Upper Beam  Shock Tube Evacuated t o about lTorr  Dump Chamber  1 He-Ne Lase:  G l a s s EndPlate -f  Front-Silvered  <4>  Mirror  Le ns  RCA 93IA Photomultiplier Scope «sLower-Beam F i g u r e 4-8  Arrangement f o r O b s e r v i n g Diaphragm Opening with Cold P r e s s u r i z e d D r i v e r  P r e s s u r e Transducer 5 V/div  Photomultiplier  20 mV/div  Diaphragm Fully Open  Time 50 usec/div  F i g u r e 4-9  140  t  r—» Diaphragm Begins to Break  P h o t o m u l t i p l i e r and Pressure Transducer Records of Diaphragm Opening  ©  Argon D r i v e r  X  Helium D r i v e r  Gas Gas  120  ftlOO •  80-  . 10  —i—  14  .12 P^*** ((pounds per square inch)  F i g u r e 4-10  Diaphragm Dynamics f o r Cold D r i v e r  Gas  °' ) 5  58 The r e s u l t s o f F i g u r e 4-10 show that the diaphragm opening time i s independent o f the c o l d d r i v e r sound speed f o r a t h r e e - f o l d change i n t h i s q u a n t i t y s i n c e argon and helium gave the same opening t i m e s .  The sound speed o f the d i s c h a r g e -  heated d r i v e r was a l s o a l t e r e d by d i l u t i n g the helium d r i v e r gas w i t h 12% a i r by volume.  (At the gap s e t t i n g used, the  d i s c h a r g e would not " f i r e " f o r a g r e a t e r amount o f  air).  I n c r e a s i n g the mass d e n s i t y of the d r i v e r i n t h i s way decreases the  sound speed by roughly 25%.  diaphragm  No s i g n i f i c a n t change i n the  opening time o c c u r r e d .  From these two p i e c e s o f  evidence i t appears t h a t the diaphragm  opening time does not  depend s t r o n g l y on the d r i v e r sound speed and so may be used as a measure o f the d r i v e r p r e s s u r e . Using  (4.5), and (4.6) and (4.7), i t i s now p o s s i b l e ,  from the c a p a c i t o r bank v o l t a g e , to c a l c u l a t e the d r i v e r gas pressure: 2 p. = 0.332 V 4  c  atmospheres (V i n k i l o v o l t s ) c  T h i s e m p i r i c a l r e s u l t i s o f course o f l i t t l e  (4.8)  significance  u n l e s s one i s sure t h a t the p r e s s u r e i n the d r i v e r i s reasonably uniform.  Two simple experiments i n d i c a t e d that t h i s  condition  i s p r o b a b l y achieved. The smear camera p i c t u r e s o f the diaphragm  opening show  t h a t both l a t e r a l p e t a l s open i n about the same time.  Rotating  the  image o f the smear-camera s l i t  top  and bottom p e t a l s showed that they opened i n the same time  as the l a t e r a l p e t a l s .  This r e s u l t  i s u n i f o r m over the d r i v e r  through 90° t o observe the  i n d i c a t e s t h a t the p r e s s u r e  cross-section.  59 Another i n d i c a t i o n of the d r i v e r u n i f o r m i t y was as f o l l o w s .  The  time between i n i t i a t i o n of the d i s c h a r g e  the i n s t a n t at which the diaphragm begins rupture time, was tube a x i s behind  obtained and  to break, the diaphragm  measured by p l a c i n g a photodiode  on the shock  a g l a s s p l a t e s e a l i n g the dump-chamber.  The  f i r s t d r i v e r l u m i n o s i t y t r a n s m i t t e d by the diaphragm a c t i v a t e d the d i o d e .  The  c u r r e n t waveform was  simultaneously  monitored.  A t y p i c a l o s c i l l o s c o p e r e c o r d i s shown i n F i g u r e 4-11. v a l i d i t y of rupture times measured i n t h i s way  was  r e c o r d i n g s i m u l t a n e o u s l y the rupture and opening smear camera. addition, light  checked by  times w i t h  The b a s i c arrangement of F i g u r e 4-4 from the d i s c h a r g e was  The  was  used.  producing  a s t r e a k on  the f i l m along s i d e the image of the diaphragm opening.  The  i n t e r v a l between the i n i t i a t i o n of the d i s c h a r g e and  breakage of the diaphragm measured i n t h i s way photodiode  measurement of the rupture time.  measurements show t h a t the d r i v e r gas  the  agreed w i t h  The  i s heated  rupture  the  time  at constant  volume s i n c e the diaphragm does not begin to break u n t i l d i s c h a r g e c u r r e n t has  the  f a l l e n to a s m a l l v a l u e .  To o b t a i n an i n d i c a t i o n of the d r i v e r gas u n i f o r m i t y , the rupture time was diaphragm f i r s t  compared w i t h the time  A pressure  a L u c i t e p l a t e s e a l i n g the d r i v e r  s e c t i o n showed that a shock wave generated reaches  at which the  f e e l s the e f f e c t of the d i s c h a r g e .  probe i n s e r t e d through  the diaphragm very q u i c k l y .  In  transmitted v i a a l i g h t  p i p e to the edge of the smear camera s l i t  time  the  The  shock i s compared w i t h the rupture time  by the  arrival  discharge time of t h i s  i n F i g u r e 4-12.  The  Time 10 p.sec/div  C u r r e n t Waveform 5 V / d i v Photodiode S i g n a l  D i s c h a r g e V o l t a g e = 10 KV Diaphragm T h i c k n e s s = 0.005 i n c h  F i g u r e 4-11  5  Diaphragm Breaks  Discharge Begins  C u r r e n t Waveform and Diaphragm Rupture Time  Time f o r D r i v e r Shock t o Reach Diaphragm [ Diaphragm Rupture Time  A O  I  I 0 F i g u r e 4-12  0.05  V~  ,  0.10 , ((KV)  0.15 )  Times C h a r a c t e r i s t i c of D i s c h a r g e and Diaphragm  0.20  61 " a r r i v a l " times times.  are mu'cli s h o r t e r than the diaphragm rupture  T h i s f a c t i n d i c a t e s t h a t there i s ample time f o r  c o n d i t i o n s i n the d r i v e r to become f a i r l y u n i f o r m b e f o r e diaphragm breaks.  I t t h e r e f o r e seems probable  t h a t the  d r i v e r u n i f o r m i t y approaches that i n c o n v e n t i o n a l d r i v e n shock  tubes.  the  pressure  62  CHAPTER '5 COMPARISON OF SHOCK TUBE OPERATION WITH IDEAL THEORY  5.1  Introduction In the l a s t c h a p t e r we showed t h a t the Smy shock tube i s  a p r e s s u r e - d r i v e n shock tube and measured t h e d r i v e r p r e s s u r e , p^ ( e q u a t i o n ( 4 . 8 ) ) .  We now c o n s i d e r the p r o b l e m o f p r e d i c t i n g  the shock speed from t h e i n i t i a l p a r a m e t e r s , V , t h e c a p a c i t o r v o l t a g e , and p^, the t e s t gas p r e s s u r e . The o p e r a t i o n o f t h e shock tube i s compared w i t h t h e i d e a l t h e o r y developed i n Chapter 2 f o r o n e - d i m e n s i o n a l gas f l o w i n t h e f o l l o w i n g manner.  The t h e o r e t i c a l dependence o f shock  speed on diaphragm p r e s s u r e r a t i o , V/^/Vi* i s c a l c u l a t e d w i t h t h e sound speed r a t i o , a^/a^, as p a r a m e t e r , and t h e o r e t i c a l c u r v e s o f shock speed v e r s u s l o g ( p ^ / p ) a r e p l o t t e d . 1 0  1  Then, f o r a  g i v e n d r i v e r v o l t a g e , shock speeds a r e measured as a f u n c t i o n o f downstream p r e s s u r e , p^.  I n order t o bring the s e t of experi-  m e n t a l p o i n t s i n t o c o i n c i d e n c e w i t h one o f t h e t h e o r e t i c a l c u r v e s , i t i s n e c e s s a r y t o assume a d r i v i n g p r e s s u r e , p^.  Unique  v a l u e s o f p | and a | a r e d e t e r m i n e d i n t h i s way f o r s e v e r a l d i f f e r e n t bank v o l t a g e s .  ( a ^ denotes t h e e s t i m a t e d v a l u e o f t h e  d r i v e r sound speed o b t a i n e d by matching t h e e x p e r i m e n t a l and t h e o r e t i c a l curves as d e s c r i b e d . ) The r e s u l t i n g e s t i m a t e s o f d r i v e r p r e s s u r e a r e t h e n compared w i t h t h e v a l u e s d e t e r m i n e d from t h e diaphragm opening times (see Chapter 4 ) . However, s i n c e t h e shock speed i s n o t c o n s t a n t a l o n g the t u b e , a s l i g h t problem a r i s e s i n s e l e c t i n g t h e speed t o be used  63 in  the above p r o c e d u r e .  it  i s unique.  of  the maximum shock speed f o r a v a r i e t y o f downstream p r e s s u r e s ,  I t was  The maximum speed was  t h e r e f o r e n e c e s s a r y to l o c a t e the p o s i t i o n  p^, and d r i v e r c o n d i t i o n s . the  chosen because  These measurements are d e s c r i b e d i n  next s e c t i o n , and i n the f o l l o w i n g s e c t i o n , the e s t i m a t e d  v a l u e s o f p^ are compared w i t h the v a l u e s o f p^ found i n Chapter 4.  5.2  F i n a l l y , the " e f f i c i e n c y " of the shock tube i s d i s c u s s e d .  Dependence o f Shock Speed on A x i a l Six  Position  p r e s s u r e probes were i n s e r t e d along the shock tube  f l u s h w i t h the i n s i d e w a l l .  T h e i r s i g n a l s were s i m u l t a n e o u s l y  r e c o r d e d on two T e k t r o n i x o s c i l l o s c o p e s , a 545A and a d u a l beam 551, u s i n g three Type 1A1 P l u g - i n U n i t s operated i n the mode to g i v e s i x c h a n n e l s .  A photodiode, a c t i v a t e d by  from the d r i v e r d i s c h a r g e when the diaphragm  chopped light  began to break,  t r i g g e r e d one o s c i l l o s c o p e and a c a l i b r a t e d d e l a y u n i t  which  t r i g g e r e d the second o s c i l l o s c o p e a t an a p p r o p r i a t e time. time lapse between probe s i g n a l s was  then used to c a l c u l a t e the  average shock speed over the i n t e r v a l between probes. shock was  The  assumed to have t h i s speed midway between the probes.  The v a r i a t i o n o f shock speed w i t h a x i a l p o s i t i o n over the 1.6  The  meter l e n g t h o f the shock tube was measured i n t h i s  first  way.  J o i n i n g the p o i n t s w i t h a smooth curve enabled a good estimate of  the maximum shock speed and i t s l o c a t i o n t o be made. The r e s u l t s of these measurements are shown i n  F i g u r e s 5 - l a and 5 - l b . discussion.  The f e a t u r e s o f these graphs m e r i t some  Over the i n i t i a l p e r i o d of i t s f l i g h t , the shock  X j  i  i  i  0  20  40  60  Figure 5-la)  i  80 Distance  i  100  i  120  (cm.)  Dependence o f Shock Speed on D i s t a n c e from the Diaphragm  i  140  i  160  66 f r o n t i s seen to a c c e l e r a t e .  A f t e r r e a c h i n g a maximum speed,  the p o s i t i o n of which depends on the downstream p r e s s u r e , the shock wave then d e c e l e r a t e s . The a c c e l e r a t i o n has been a t t r i b u t e d to e f f e c t s of the diaphragm  opening^  24  » ).  In the i d e a l t h e o r y , the diaphragm  25  i s assumed to d i s a p p e a r i n s t a n t a n e o u s l y and the flow i s assumed to be one-dimensional.  The shock wave then maintains a w e l l -  d e f i n e d speed determined by the p r e s s u r e r a t i o , p^/p^, and the sound speed r a t i o , a^/a^.  In r e a l i t y , however, the diaphragm  opening process r e q u i r e s a f i n i t e time, d u r i n g which the d r i v e r gas i s e x p e l l e d through an aperture of i n c r e a s i n g s i z e at a r a t e determined by P^/p^ and a^. d i m e n s i o n a l because flow v e l o c i t y .  initially  Furthermore, the flow i s t h r e e there are r a d i a l components i n the  D r i v e r gas passes through the a p e r t u r e at a  p r e s s u r e determined by i t s r a t e o f e x p u l s i o n , and expands t o fill  the tube c r o s s - s e c t i o n .  As the diaphragm  s i z e , the e x p e l l e d d r i v e r gas need expand so i t s n e t drop i n p r e s s u r e i s l e s s . the flow v e l o c i t y i s a l s o reduced.  opening grows i n  less to f i l l  the.tube,  The r a d i a l component of The r e s u l t i n g p r e s s u r e  g r a d i e n t i n the e x p e l l e d d r i v e r gas a c c e l e r a t e s the shock. The d e c e l e r a t i o n of the shock i s due t o s e v e r a l phenomena. V i s c o u s and thermal boundary  l a y e r s near the w a l l o f the shock  tube cause the shock wave to attenuate as d i s c u s s e d i n S e c t i o n 2.7.  Another e f f e c t can a l s o i n f l u e n c e the shock speed.  When  the diaphragm b r e a k s , a r a r e f a c t i o n wave t r a v e l s backwards through the d r i v e r gas, i s r e f l e c t e d at the end w a l l o f the d r i v e r and then proceeds down the shock tube i n p u r s u i t of the  67 shock f r o n t .  When the r e f l e c t e d r a r e f a c t i o n wave overtakes the  c o n t a c t s u r f a c e , the d r i v e r gas p r e s s u r e  i s reduced below the  v a l u e r e q u i r e d to keep the shock heated gas moving at constant speed ( i . e .  drops below P ) •  s l o w l y at f i r s t  2  The shock speed then drops,  s i n c e the v e l o c i t y and. pressure  gradients i n  the r e f l e c t e d r a r e f a c t i o n wave are u s u a l l y s m a l l p r o v i d e d the o v e r t a k i n g does not occur too c l o s e t o the d r i v e r s e c t i o n . constant  d r i v e r c o n d i t i o n s , the d i s t a n c e from the diaphragm at  which t h i s occurs process for  For  probably  i s s h o r t e r the lower the shock speed. accounts f o r the l a r g e r a t t e n u a t i o n  shocks at t e s t gas p r e s s u r e s  Below 5 T o r r , the above process beginning  above about 5 T o r r  This  observed ( F i g u r e 5-1).  e i t h e r d i d not occur or was j u s t  t o occur i n the length o f shock tube s t u d i e d .  In a d d i t i o n , c o o l i n g of the hot d r i v e r gas as i t flows down the tube may reduce i t s pressure  and sound speed and con-  t r i b u t e t o the a t t e n u a t i o n . I f the i n i t i a l opening process  shock a c c e l e r a t i o n i s due to the diaphragm  as d e s c r i b e d , then one would expect the d i s t a n c e ,  L, over which the shock a c c e l e r a t e s to be o f the order o f the d i s t a n c e t r a v e l l e d by the shock f r o n t d u r i n g the opening t  : i.e. op  L  ~ vmax ' top  where v i s the maximum shock speed. max r  time,  (5.1)  v  The measurements o f  F i g u r e 5-1, made w i t h diaphragms 0.005 i n c h t h i c k and a bank voltage  of 10 KV, support  t h i s e x p e c t a t i o n and give  L = 1.3v t max op m Q v  '  (5.2)  68 F i g u r e 5-2  shows t h a t an i n c r e a s e i n t  ,. caused by i n c r e a s i n g  the diaphragm t h i c k n e s s while m a i n t a i n i n g at  10 KV,  r e s u l t s i n an i n c r e a s e i n L.  Moreover, the  i n c r e a s e i n L i s c o n s i s t e n t w i t h equation F i g u r e 5-3  the c a p a c i t o r v o l t a g e resulting  (5.2), as shown i n  where the measurements f o r both 0.005 i n c h and  i n c h t h i c k diaphragms are seen to f i t the same s t r a i g h t The  line.  dependence of L on d r i v e r gas p r e s s u r e , p^,  checked a l s o .  The  d r i v e r p r e s s u r e was  reduced from 33  0.010  was atmospheres,  as i n the above measurements, to 16 atmospheres by d e c r e a s i n g initial  helium  d r i v e r gas p r e s s u r e  from 1 atmosphere t o 100  keeping the c a p a c i t o r .voltage at 10 KV.  The  results  the  Torr,  (Figure  5-4)  i n d i c a t e t h a t the p o s i t i o n of maximum shock speed, L, i s independent of the d r i v e r p r e s s u r e ,  p^.  These measurements s t r o n g l y suggest t h a t the  initial  a c c e l e r a t i o n of the shock wave i s a t t r i b u t a b l e to the time r e q u i r e d f o r the diaphragm to open c o m p l e t e l y , q u a l i t a t i v e agreement with the r e s u l t s quoted by et  5.3  al/  2  5  finite  and  are i n  Simpson  '  Comparison of Shock Tube Operation w i t h I t i s now  Ideal  Operation  p o s s i b l e t o compare the observed maximum shock  speeds w i t h the speeds p r e d i c t e d by the theory d i s c u s s e d i n Chapter 2 ( S e c t i o n s 2.3 We  and  d e f i n e p^ as the p r e s s u r e  admitting  t h a t p | may  differ  the diaphragm opening times. below.  2.4),  as o u t l i n e d i n S e c t i o n  5.1.  e f f e c t i v e i n d r i v i n g the shock, from the p r e s s u r e The  meaning o f p^  obtained  from  is clarified  Having determined the maximum shock speed as a f u n c t i o n  o —  1  0 Figure  1  1  40 5-2  Dependence  1 80 Distance (cm.)  of Shock  Speed  on P o s i t i o n  j  1  1  120 for Different  I  160 Diaphragm  Thicknesses  100 "  0 F i g u r e 5-3  10  2 0 - 3 0  40 . ^ax^op  Dependence o f L on t h e P r o d u c t v  50 (cm  -> *t  60  70  ©  D r i v e r P r e s s u r e = 33 a t m o s p h e r e s D i a p h r a g m O p e n i n g T i m e = 100 \isec  A  D r i v e r P r e s s u r e = 16 a t m o s p h e r e s D i a p h r a g m O p e n i n g Time = 150 [isec Diaphragm  1 0  1  1  40 Figure  5-4  Dependence o f Shock  1  , 8 0 Distance (cm.) Speed  1  Thickness  = 0.005  1 120  on P o s i t i o n f o r D i f f e r e n t D r i v e r  i  inch  1— 160 Conditions  72 of t e s t gas p r e s s u r e , p^, f o r constant c a p a c i t o r bank v o l t a g e , we  then p l o t the maximum Mach number a g a i n s t log^QCp^/p^)  assuming i n i t i a l l y  that p | i s 1 atmosphere.  The r e s u l t i n g  i s then compared w i t h t h e o r e t i c a l curves c a l c u l a t e d from theory of S e c t i o n s 2.3  and  2.4.  curve  the  A v a r i e t y of such curves f o r  d i f f e r e n t assumed values of  and a| (the e f f e c t i v e  gas  The experimental curve i s then  sound speed) are p l o t t e d .  driver  s h i f t e d along the l o g ( p | / p ) - a x i s u n t i l i t c o i n c i d e s w i t h 1 0  of  1  the t h e o r e t i c a l c u r v e s .  The values of p^ and  c h a r a c t e r i z i n g t h i s t h e o r e t i c a l curve we  one  a^  then c a l l  the  effective  d r i v e r gas p r e s s u r e and sound speed r e s p e c t i v e l y , because they are the v a l u e s which we must s u b s t i t u t e i n t o the theory i n order to p r e d i c t the observed of  shock speeds.  The a n t i l o g a r i t h m  t h i s s h i f t gives the e f f e c t i v e d r i v e r gas p r e s s u r e ,  atmospheres.  T h i s procedure  i s shown i n F i g u r e s 5-5  , in  to 5-7 f o r  s e v e r a l c a p a c i t o r bank v o l t a g e s and the r e s u l t s are c o l l e c t e d i n Table I below. Bank Voltage  Pi  Effective Sound Speed a«  (KV)  (atm)  (m/sec)  7  5.3  2750  16.9  0.59  10  10.5  3500  33.2  0.67  14  15.1  4250  66.3  0.64  V  c  Effective Driver Pressure  Table I  «  Driver Pressure p  l o  Sin  ^£ p.  4  (atm)  Shock Tube D r i v e r C o n d i t i o n s  73  16.6  13.3  Bank V o l t a g e = 14 KV © Experimental Points assuming p^ = 1 atmosphere 23  X Shifted Points Shift  9.95  ©  p' = Assumea D r i v e r ^ Pressure i n Atmospheres  21  P  = T e s t Gas Pressure  1  Theory of Chapter 2  19  17 u Xt  H  CO  s  o o  X! Kl  13 -  11 -  0  1  2  3 log (p^/p ) F i g u r e 5-5 Measured Shock Speed v e r s u s 1 Q  4  1  log^ (p^/p^) 0  F i g u r e 5-6 Measured Shock Speed v e r s u s l o g  (p'/p,)  75  l o g  10  ( p  4  / p  ) 1  J i g u r e 5-7 Measured Shock Speed v e r s u s  log (p^/p ) 1 Q  1  76 The v a l u e s o f amounts.  and p^ are seen t o d i f f e r by l a r g e  Since the t h e o r y f o r the diaphragm  opening has been  v e r i f i e d under predetermined d r i v e r c o n d i t i o n s , i t seems likely  that the diaphragm  opening times should g i v e r e l i a b l e  v a l u e s f o r the d r i v e r p r e s s u r e , p^.  The f a c t t h a t the e f f e c t i v e  d r i v i n g p r e s s u r e , p^, i s c o n s i d e r a b l y l e s s than p^ i s p r o b a b l y due  t o the t h r e e - d i m e n s i o n a l flow p a t t e r n of the d r i v e r gas as  i t passes through the diaphragm  opening.  Consider a snapshot  o f the flow taken at a time when the diaphragm  has opened  through a s m a l l angle 6 (Figure 5-8). The arrows d i r e c t i o n o f gas flow;  F i g u r e 5-8 initially  i n d i c a t e the  I t i s e v i d e n t that the d r i v e r gas  I n i t i a l D r i v e r Gas Flow  Pattern  e x p e l l e d through the s m a l l opening w i l l have a l a r g e  r a d i a l component o f momentum.  The q u a n t i t y of d r i v e r gas  which passes through the diaphragm  d u r i n g the opening process  and which consequently executes t h i s r a d i a l motion can be estimated from the mean diaphragm  a p e r t u r e d u r i n g the opening  p r o c e s s and the e x p u l s i o n speed of the d r i v e r gas. The l a t t e r can be c a l c u l a t e d from the shock speed near the diaphragm. Such a c a l c u l a t i o n shows that a l a r g e f r a c t i o n o f the d r i v e r  gas  i s e x p e l l e d during the opening time and must be a f f e c t e d by-  three-dimensional  motion.  motion i s probably  The  energy i n v o l v e d i n t h i s  l a r g e l y absorbed by the shock tube w a l l s  so i s i n e f f e c t i v e i n d r i v i n g the shock. a x i a l momentum t r a n s f e r to the t e s t gas  i s much l e s s than The  r e s u l t s presented  i n the t a b l e above show t h a t p^  by Pj = B(a|)  . (5.4)  2  -2 where B i s a constant equal to 8.1(±0.8)*10 T h e o r e t i c a l l y , one would expect  ^  the d r i v e r gas.  -3 Kg-m  that  V\ - - J - Cap and  reduced  observe.  and a^ are r e l a t e d  where  i n the  shocks  should behave then as though d r i v e n by a c o n s i d e r a b l y  The  and  Consequently, the  i d e a l s i t u a t i o n , where o n l y a x i a l flow o c c u r s .  p r e s s u r e , as we  radial  (S.S)  2  are the mass d e n s i t y and a d i a b a t i c exponent of Assuming  ^  = 1.67,  one  finds  fit- = 9.8-10" Kg.m" , 2  i n reasonably  3  good agreement with the value of B  found  experimentally. I t t h e r e f o r e seems, i n s p i t e of the i n i t i a l dimensional  d r i v e r gas  dimensional  theory can be  p r o v i d e d we  use  three-  flow p a t t e r n , t h a t the i d e a l a p p l i e d to a good  one-  approximation  an e f f e c t i v e d r i v e r gas p r e s s u r e , p^, and  an  1/2 e f f e c t i v e d r i v e r gas sound speed, a| = ( 3^4P4/ JP 4^ E m p i r i c a l l y we f i n d t h a t  PA >4 » ( P J  S  C5.6)  78 f o r 15 atm.  < p  are expressed C4.8)  < 70 atm.,  4  where s = 0.63  i n atmospheres.  I t i s now  ± 6%,  4  and  p^  p o s s i b l e , u s i n g equation  f o r p^ and the above r e l a t i o n s f o r p | and  shock speeds i n the Smy  and p  a^, to p r e d i c t  shock tube by means of the  standard  shock tube theory knowing only the c a p a c i t o r bank v o l t a g e , V  ,'  the t e s t gas p r e s s u r e , p^, and the d r i v e r gas mass d e n s i t y ,  J>^>  which may  be c a l c u l a t e d from the d r i v e r f i l l i n g p r e s s u r e s i n c e  the d r i v e r gas  i s heated  I t has  at constant volume.  s i m i l a r l y been observed  that even f o r c o l d - d r i v e r  diaphragm shock tubes the i d e a l theory does not and  t h a t an e f f e c t i v e d r i v i n g p r e s s u r e has t o be i n t r o d u c e d to  p r e d i c t the Mach numbers. to a v a r i a t i o n of  $^  w i t h temperature  T h i s d i s c r e p a n c y has been a t t r i b u t e d  (the a d i a b a t i c exponent of the d r i v e r  as the d r i v e r gas c o o l s through  wave (see S e c t i o n 2.4).  We  apply i n our case s i n c e the all  apply  temperatures  gas temperature breaks  the  gas)  expansion  f e e l that t h i s e x p l a n a t i o n does not )f f o r helium i s almost  below about 1.5 • l O ^ K ^  3 5  constant f o r  ^ , and s i n c e the  driver  i s l e s s than t h i s value when the diaphragm  f o r bank v o l t a g e s l e s s than 12 KV.  motion of the expanding  d r i v e r gas  f o r the d i s c r e p a n c i e s we  The  three-dimensional  i s then probably r e s p o n s i b l e  observe.  I t i s i n t e r e s t i n g t o note t h a t the exponent i n equation (5.6)  i s very n e a r l y  f a c t suggests a low value of  t h a t one Jf  4  ^  - 1 f o r the helium d r i v e r gas.  This  should t r y a polyatomic d r i v e r gas  to see i f the exponent i s indeed  If.  with  - 1.  79 5.4  The E f f i c i e n c y o f the Shock: Tube The f r a c t i o n of the i n i t i a l bank energy d i s s i p a t e d i n  the d r i v e r gas was measured as f o l l o w s .  A T e k t r o n i x P6013  1000:1 v o l t a g e d i v i d e r was used to measure the v o l t a g e across the d r i v e r e l e c t r o d e s ,  .  The d i s c h a r g e c u r r e n t , I , was a l s o  measured under the same c o n d i t i o n s . I'VJJ  was  The power  dissipation,  c a l c u l a t e d and p l o t t e d as a f u n c t i o n of time.  The area  under the graph was measured w i t h a p l a n i m e t e r t o g i v e the energy d i s s i p a t i o n , W = J'l'V^dt.  The e l e c t r i c a l e f f i c i e n c y i s  d e f i n e d as  £CV and  CS.7)  2 C  the r e s u l t s are g i v e n i n Table I I .  W  1/2CV c (Joules)  (Joules)  7  1225  610  501  10  2550  1360  54%  14  4900  2540  51%  2  c (KV) V  Table I I E l e c t r i c a l E f f i c i e n c y o f the Shock Tube An e l e c t r i c a l e f f i c i e n c y of about 50% i s expected  since  the bank energy should be d i v i d e d roughly e q u a l l y between the d r i v e r spark and the a i r t r i g g e r spark.  Since  this  i n e f f i c i e n c y can be l a r g e l y avoided by a l t e r i n g the d e s i g n of the d r i v e r s e c t i o n to i n c l u d e a t r i g g e r i n g not be f u r t h e r  considered.  facility,  i t will  80 Assuming t h a t the energy, W, a l l appears as thermal energy o f p a r t i c l e motion i n the d r i v e r gas, one can c a l c u l a t e an i d e a l p r e s s u r e , p^-, f o r the d r i v e r gas. efficiency,  A thermal  7| , can then be d e f i n e d as T  ^  T  - P /P 4  (5.8)  T  where p j = W/(Volume o f d r i v e r gas). and p  4  i s the d r i v e r p r e s s u r e determined from the diaphragm  opening time.  R e s u l t s of these c a l c u l a t i o n s are p r e s e n t e d ' i n 3  ,Table I I I f o r the d r i v e r gas volume o f 167 cm  i n i t i a l l y at  atmospheric p r e s s u r e and room temperature.  c (KV) V  7 10 14  Pi (atm)  P4 (atm)  36.2  16.9  471  80.5  33.2  411  66.3  44%  151  Table I I I Thermal E f f i c i e n c y o f the Shock Tube Thus, about 45% o f the energy d i s s i p a t e d i n the d r i v e r i s a c t u a l l y e f f e c t i v e i n r a i s i n g the d r i v e r p r e s s u r e .  The r e s t  o f the energy may be l o s t i n e v a p o r a t i n g e l e c t r o d e and w a l l m a t e r i a l s and i n h e a t i n g the e l e c t r o d e s which have a l a r g e surface area.  I t t h e r e f o r e appears t h a t about 25% o f the  e l e c t r i c a l energy s t o r e d i n the c a p a c i t o r bank i s e f f e c t i v e i n h e a t i n g the d r i v e r gas t o a p r e s s u r e p..  CHAPTER 6  6.1  THE SHOCK-HEATED GAS  Introduction The u s e f u l n e s s o f a shock tube i s dependent  upon i t s  a b i l i t y to produce a u n i f o r m sample of hot gas v/ith p r o p e r t i e s c a l c u l a b l e from the i n i t i a l t e s t gas c o n d i t i o n s and the shock speed.  E x p e r i e n c e has shown, i n the case o f e l e c t r o m a g n e t i c a l l y  d r i v e n shock tubes, that a cursory examination o f the shock speed i s i n s u f f i c i e n t to e s t a b l i s h these c o n d i t i o n s t * ^ . 3  the  development  in  In  o f a shock tube, i t i s t h e r e f o r e e s s e n t i a l t o  s c r u t i n i z e i t s performance In  4  in detail.  Chapters 4 and 5, we have shown t h a t the shock speed  the Smy shock tube can be p r e d i c t e d from the independent  i n i t i a l parameters, p^, p , and V" . 4  c  We now t u r n our a t t e n t i o n  to  the most important q u e s t i o n o f the q u a n t i t y and p r o p e r t i e s  of  the shock-heated gas. Frequent r e f e r e n c e w i l l be made t o the r e g i o n s of gas  flow i n the shock tube.  These are i l l u s t r a t e d i n F i g u r e 6-1,  and a d e s c r i p t i o n of them i s given i n Chapter 2. of the  The r e g i o n  gas between the shock f r o n t and the c o n t a c t s u r f a c e i s c a l l e d shock-heated gas even though i t s p r o p e r t i e s have not y e t  been shown t o conform to the p r e d i c t i o n s o f shock wave t h e o r y . F i g u r e 6-2 i s a smear camera p i c t u r e taken w i t h the s l i t p e r p e n d i c u l a r to the shock tube a x i s .  The shock  front,  r e l a x a t i o n zone, shock-heated r e g i o n , and c o n t a c t zone are c l e a r l y seen.  ( I n p r a c t i c e , the t e s t and d r i v e r gases are not  s h a r p l y separated but tend t o mix w i t h each other over an  Wall  Test  Boundary Layer  Driver  Shock-Heated  Gas  Gas  1  2  Gas 3 X.^---——  Shock Front  Contact  Onset o f Equilibrium Conditions Relaxation Zone  Zone  F i g u r e 6-1 A x i a l S e c t i o n o f Shock Tube Showing Regions of Gas Flow  |L  Shock Front H o r i z o n t a l Scale:  J„  Shock-Heated Gas  Contact  Zone  1 cm. on f i l m = 3 . 6 cm. i n shock tube  F i g u r e 6-2 Smear P i c t u r e o f Shock Taken w i t h . S l i t P e r p e n d i c u l a r t o Shock Tube A x i s (p =10 T o r r , V =10 KV, x=120 cm) 1  c  83 extended r e g i o n which we  call  the c o n t a c t  zone.  The  diffuse,  somewhat i r r e g u l a r nature o f t h i s zone i s e v i d e n t i n F i g u r e The l e a d i n g edge of t h i s zone we contact which at for  zone was  call  the c o n t a c t s u r f a c e . )  The  i d e n t i f i e d by a shock r e f l e c t i o n technique  i s d e s c r i b e d i n the next s e c t i o n .  the shock f r o n t i s presumably  The  luminous  line  seen  due t o e a s i l y e x c i t e d i m p u r i t i e s  which the r e l a x a t i o n times are s h o r t (  luminous  6-2.  1 0  >  1 4  »1 » 9  2 9  »  5 2  )  #  T  h  i  s  l i n e shows t h a t the shock f r o n t i s p l a n a r over at  l e a s t the c e n t r a l 801 of the tube diameter. In S e c t i o n 6.2,  the q u a n t i t y of shock-heated gas i s  i n v e s t i g a t e d and accounted f o r i n terms o f a simple model. p r o p e r t i e s of t h i s gas are d i s c u s s e d i n S e c t i o n  6.2  Extent  of the Region of Shock-Heated  ^/f  >  {  t  n  e  6.4.  Gas  I d e a l l y , f o r a shock wave of constant density r a t i o ,  The  speed, V  r e g i o n of shock-heated gas  , and should  have a t h i c k n e s s ;'-ltw at  =  *  /f  P  a d i s t a n c e x from the diaphragm.  contact surface, V  (6.1)  2  A l s o , the speed of the  , should be g i v e n by  V«  =  V , ( l -  % )  (6.2)  These equations f o l l o w from the requirement of mass c o n s e r v a t i o n . From (6.2) the t h i c k n e s s of shock-heated gas should  i n c r e a s e at  the i d e a l r a t e  _ \/ _ w  d  t  -  V -V S  C  _ 5  as the shock t r a v e l s down the tube. expressions,  =  \/V . - A £-  (-3)  In d e r i v i n g these  i t i s assumed t h a t a l l of the t e s t gas encountered  by the shock i s compressed  i n t o a l a y e r between the shock f r o n t  and c o n t a c t s u r f a c e , and f o l l o w s the shock at a speed V  .  If  t h i s assumption were v a l i d , then a measurement o f the shockheated gas t h i c k n e s s or of the r e l a t i v e v e l o c i t i e s of shock f r o n t and contact s u r f a c e would serve t o determine the shock density  ratio, In r e a l i t y , however, l o s s mechanisms reduce the  o f the shock-heated l a y e r .  thickness  The mixing of t e s t and d r i v e r gases  i n the contact zone e f f e c t i v e l y advances  the c o n t a c t  surface  (the l e a d i n g edge of t h i s zone) towards  the shock f r o n t .  the establishment  l a y e r at the shock tube  of a v i s c o u s boundary  Also,  w a l l r e q u i r e s t h a t gas near the w a l l have a low v e l o c i t y . can then be passed over by the c o n t a c t s u r f a c e .  It  Furthermore,  the temperature o f gas near the w a l l must be t h a t of the w a l l . Since i t i s assumed that no pressure boundary  g r a d i e n t e x i s t s across the  l a y e r , the gas d e n s i t y near the w a l l can be much h i g h e r  (311 than near the axis'-  '.  A l a r g e f r a c t i o n o f the shock-heated  gas can thus be removed by the boundary  layer.  These  processes  reduce the t h i c k n e s s of shock-heated gas and i n c r e a s e the v e l o c i t y o f the c o n t a c t s u r f a c e . thickness, l , m  ratio, R , L  Consequently, the observed  o f shock-heated gas gives an apparent d e n s i t y  such t h a t  l  m  - X/R  (6.4)  L  i f the l o s s e s are not taken i n t o c o n s i d e r a t i o n . l e s s than 1 ^  and R  theoretical value,  L  correspondingly P ^/p> .  the c o n t a c t s u r f a c e speed, V  QSm  will  be  g r e a t e r than the  In a s i m i l a r way,  2  1  a measurement of  and shock speed, V , g  gives  an  85 apparent d e n s i t y r a t i o , R^, r e l a t e d to the r a t e o f i n c r e a s e o f 1 m (6.5)  csm Now, due t o the l o s s of shock-heated gas, V  the  instantaneous r a t e o f l o s s .  the  boundary  w i l l be g r e a t e r  Although the l o s s of gas through  l a y e r and through mixing reduces l  m  , these p r o c e s s e s  do not a f f e c t the p r o p e r t i e s o f the shock-heated gas and so can be t o l e r a t e d p r o v i d e d t h a t a l l o f the gas i s not removed. In  a d d i t i o n to the l o s s mechanisms, another phenomenon  may a f f e c t the observed l e n g t h o f shock-heated of  increase.  When the diaphragm  breaks and d r i v e r gas rushes  downstream, a r a r e f a c t i o n wave t r a v e l s towards of the  gas and i t s r a t e  the d r i v e r s e c t i o n where i t i s r e f l e c t e d .  the c l o s e d end After  reflection,  r a r e f a c t i o n wave proceeds downstream i n p u r s u i t o f the shock  f r o n t at a speed g i v e n by the sunt o f the l o c a l flow v e l o c i t y and the  l o c a l speed o f sound.  In r e g i o n 5 (see F i g u r e 6-1) behind  the  c o n t a c t s u r f a c e , t h i s wave has a speed v^ + a^.  c o n t a c t s u r f a c e has a speed v ^ , i t w i l l the  S i n c e the  c l e a r l y be overtaken by  r a r e f a c t i o n wave a t some p o i n t i n a s u f f i c i e n t l y long shock  tube.  The p o i n t a t which  t h i s occurs depends on the i n i t i a l  c o n d i t i o n s , the shock speed and the l e n g t h o f the d r i v e r The c o n t a c t s u r f a c e w i l l shock-heated  layer w i l l  then be r e t a r d e d .  section.  As a r e s u l t , the  increase i n thickness at a rate greater  86 than p r e d i c t e d by (6.3) , and the apparent d e n s i t y r a t i o , R^., will  f a l l below the t h e o r e t i c a l v a l u e ,  &/p  »  A l s o , the  r a r e f a c t i o n wave w i l l pass i n t o the shock-heated gas C  r e  gi°  n  2  of F i g u r e 6-1) and a l t e r the p r o p e r t i e s o f t h i s g a s . C l e a r l y t h i s e f f e c t i s undesirable  and should be i n v e s t i g a t e d i n  examining the m e r i t s o f a shock tube. A comparison o f the measured v a l u e s the i d e a l d e n s i t y r a t i o ,  *^ /p a  » should  of  and  with  then g i v e an i n d i c a t i o n  of the importance o f l o s s mechanisms and o f the e f f e c t o f the r a r e f a c t i o n wave. To  i n v e s t i g a t e the extent  and growth o f the shock-heated  r e g i o n a s i x - f o o t l e n g t h o f pyrex pipe was used i n the shock tube.  The gas flow between p o s i t i o n s 100 cm and 130 cm from  the diaphragm was observed with the smear camera. graphs  ( F i g u r e 6-3) p r e s e n t  x - t diagrams of the luminous  f e a t u r e s o f the flow i n the shock tube. on the f i l m r e p r e s e n t i n g the progress be c a l l e d t r a j e c t o r i e s .  The photo-  The o b l i q u e  o f these  "streaks"  features  will  The v e r t i c a l b l a c k l i n e s are markers  spaced at 10 cm i n t e r v a l s along the shock tube.  In order t o  i d e n t i f y the contact s u r f a c e and, at the same time, to render weak, non-luminous shocks v i s i b l e , the gas flow was r e f l e c t e d from a l u c i t e d i s c f i t t e d  f l u s h w i t h the i n s i d e s u r f a c e o f the  shock tube and p e r p e n d i c u l a r downstream.  to i t s a x i s a t a p o s i t i o n 123 cm.  The t r a j e c t o r y o f the c o n t a c t s u r f a c e c o u l d be  i d e n t i f i e d by the change i n speed, or r e f r a c t i o n , of the (2 30") r e f l e c t e d shock as i t passed through the c o n t a c t  zone*- *  J  .  To reduce the i n t e n s i t y o f the l i g h t emitted by the doubly-  Neutral Density F i l t e r Covers Upper P a r t of F i l m  T e s t Gas  (a)  V:  Pressure  Denotes V e r t i c a l (Time) S c a l e on F i l m L o c a t i o n of N e u t r a l D e n s i t y F i l t e r F i g u r e 6-3  (b)  p  1  V:  = 0.5 T o r r 15 jisec/cm  Smear Photographs of Gas Flow i n Shock Tube H o r i z o n t a l Scale: 1 cm on f i l m = 5.75 cm i n shock tube a x i a l  direction  (c)  p  1  V:  = 1 Torr 15 (isec/cm  (d)  J  9  1  V:  = 2 Torr 15 u.sec/cm  =7.5  Torr  30 (j.sec/cm  (h)  p V:  = 10 T o r r 30 usec/cm  90  x Regions of Gas Flow F i g u r e 6-4  +  p  1 2 3 5  Thickness of Shock-Heated Gas  "j  P o s i t i o n of Reflecting Plate  T e s t Gas i n I n i t i a l S t a t e Shock-Heated T e s t Gas D r i v e r - T e s t Gas M i x t u r e Doubly-Shocked T e s t Gas  E x p l a n a t i o n o f the F e a t u r e s of the Smear Camera Photographs o f F i g u r e 6-3  91 shocked gas ( r e g i o n 5 of F i g u r e  6-4) i t was n e c e s s a r y to p l a c e  a n e u t r a l d e n s i t y f i l t e r d i a g o n a l l y across f i l m as shown i n F i g u r e  the photographic  6-3a. T h i s e x p l a i n s the abrupt  "fading"  of the luminous f e a t u r e s a s s o c i a t e d w i t h the i n c i d e n t shock as they approach the r e f l e c t i n g p l a t e .  A l i n e diagram ( F i g u r e 6-4)  e x p l a i n s the r e l e v a n t f e a t u r e s of a t y p i c a l smear p i c t u r e . The contact  sharpness o f the shock r e f r a c t i o n suggests t h a t the  zone i s f a i r l y t h i n .  Colour  smear p i c t u r e s a l s o e x h i b i t  a marked c o l o u r change i n the c o n t a c t r e g i o n .  I t appears  there-  f o r e t h a t a w e l l - d e f i n e d r e g i o n o f shock-heated gas e x i s t s . From the smear p i c t u r e s i t i s then p o s s i b l e t o measure the thickness  of shock-heated gas,  l  m  , ( F i g u r e 6-5), as w e l l as the  v e l o c i t i e s o f both the shock f r o n t and the c o n t a c t The  smear p i c t u r e s a l s o show that f o r  < 5 T o r r , the l e n g t h  o f the r e l a x a t i o n zone i s small compared w i t h 1^. of t h i s c o n d i t i o n i s d i s c u s s e d  d r i v e r con-  The shock speed was v a r i e d by changing the t e s t gas  p r e s s u r e , p^.  The dependence of the shock speed on p^ f o r the  d r i v e r c o n d i t i o n s used (V The  The importance  i n S e c t i o n 2.6.  A l l measurements were made w i t h constant ditions .  surface.  r e s u l t s are presented  = 10 KV) i s shown i n F i g u r e 6-5. below and then an attempt i s made t o  account f o r them q u a n t i t a t i v e l y . In Figure  6-6, the r a t i o , Ijj/lj-h* of measured to  t h e o r e t i c a l thickness  of shock-heated gas i s p l o t t e d as a  f u n c t i o n o f downstream p r e s s u r e , p^. is  l e s s than u n i t y .  exist.  In a l l cases,  this ratio •  Two p o s s i b l e reasons f o r t h i s d e v i a t i o n  E i t h e r the d e n s i t y r a t i o across  the shock i s not the  92  20  18  16  14 M  12 -  10 -  8  P F i g u r e 6-5  1  (Torr)  Dependence o f Shock Mach Number (M) and ShockHeated Gas T h i c k n e s s ( l ) on T e s t Gas P r e s s u r e (p-^) m  1.0 Bank V o l t a g e = 10 KV Measuring S t a t i o n 120 cm. from Diaphragm 0.8J in  "th 0.6 Theory (Equations (6.13) and (6,16)) 0.4 0  Experiment  0.2  0  —r  i  0 P  x  8 (Torr)  10  12  F i g u r e 6-6 Comparison of Measured T h i c k n e s s of Shock-Heated Gas Value ( l ) t  h  14 ( l ) with m  Ideal  CD  •'w  value,  P2/P],  calculating  »' determined from the shock speed (and used i n 1 ^ ) ,  o r some o f the t e s t gas has been l o s t from the  shock-heated l a y e r .  Now, i n view o f experiments d e s c r i b e d i n  S e c t i o n 6.4, i t appears t h a t the shock d e n s i t y r a t i o agrees w i t h the c a l c u l a t e d v a l u e . •  .  The d i f f e r e n c e between 1  •  and 1., must m  tn  then be due t o gas l o s s . The measured c o n t a c t s u r f a c e speed, V" . i s plotted ' csm* a g a i n s t the measured shock speed, V , i n F i g u r e 6-7. A l s o shown r  are the l i n e V V . s  csm  = V  s  r  and the t h e o r e t i c a l dependence o f V" cs r  on  The measurements are seen to d i f f e r from the p r e d i c t e d  values  f o r both h i g h shock speeds  (high p-.).  The approach o f V  (low p^) and low shock to V  speeds  i s attributed to loss of  gas from the shock-heated l a y e r s i n c e i t seems u n l i k e l y t h a t the d e n s i t y r a t i o should d i f f e r much from the t h e o r e t i c a l The d i s c r e p a n c y probably  f o r low V  s  value.  ( i . e . p.^ g r e a t e r than about 4 T o r r ) i s  due to the r e f l e c t e d ' r a r e f a c t i o n wave o v e r t a k i n g and  r e t a r d i n g the c o n t a c t s u r f a c e . explanation  i s contained  g r e a t e r than 5 T o r r . trajectories  Evidence i n support  of t h i s  i n some smear p i c t u r e s taken f o r  In some o f these photographs, subsequent  i n r e g i o n 3 ( F i g u r e 6-4) have i n c r e a s i n g slopes and  therefore represent  flow o f d e c r e a s i n g  speed.  A rarefaction  wave advancing towards the c o n t a c t s u r f a c e would produce  this  effect. The observed d e n s i t y r a t i o s R  L  and Ry, obtained  from  measurements o f the t h i c k n e s s of shock-heated gas and i t s r a t e o f i n c r e a s e r e s p e c t i v e l y (equations  (6.4) and ( 6 . 5 ) ) , are com-  pared w i t h the i d e a l v a l u e , p / p , , i n F i g u r e 6-8. 2  Over the  Bank V o l t a g e » 10 KV Measuring  S t a t i o n 120 cm. from Diaphragm  «5  Figure 6-7  C o n t a c t S u r f a c e Speed (V  ) as a F u n c t i o n o f Shock Speed  (V ) s  Bank V o l t a g e  25'  Measuring  P, 20-  =  10  KV  S t a t i o n 120  cm.  from  Diaphragm  Theoretical Density Ratio (Calculated from Measured Shock Speed) Apparent Density R a t i o ( C a l c u l a t e d from L e n g t h o f S h o c k - H e a t e d Gas Layer)  o  Apparent Density Ratio (Calcu ated Rate of Increase of Shock-Heated Gas Thickness)  •rl +->  15-  from  ei  to  >> 4->  •rl  ID  a  10-  Q  0-5 0  1  1  2  4  1 .6 Test  Figure  6-8  Comparison of  1 Gas  Density  8 Pressure Ratios  ' 10 (Torr)  ' 12  1  14  ' 16  whole range o f p r e s s u r e s  investigated,  ML  >  9x  /p  C6.6)  l  Assuming that the shock-heated gas kas d e n s i t y , implies  p , this result 2  that some o f the downstream gas encountered by the  shock f r o n t has been l o s t and does not appear i n the shockheated l a y e r .  Furthermore, the graph a l s o shows t h a t  for a l l  p r e s s u r e s used -  R  L  > Ry  (6.7)  Hence, gas i s l o s t e a r l y i n the f l i g h t o f the shock a t a r a t e g r e a t e r than the l o s s r a t e from the diaphragm.  at the o b s e r v a t i o n s t a t i o n 120 cm  The i n i t i a l h i g h l o s s o f gas i s probably  due  to mixing of t e s t and d r i v e r gases during the f o r m a t i o n o f  the  shock f r o n t .  A further  feature  of.Figure  6-8 deserves  comment b e f o r e a model f o r the gas flow i s d i s c u s s e d . regimes can be d i s t i n g u i s h e d •  For  by the r e l a t i v e s i z e s o f Ry and  l e s s than about 4 T o r r , Ry  explained i n discussing  > Pa^, . As  equation (6.5), t h i s r e s u l t s from t h e  l o s s o f t e s t gas, probably through a v i s c o u s boundary For p^ > 4 T o r r ,  layer.  Ry < ij/j? ^ i n d i c a t i n g that the t h i c k n e s s o f  shock-heated gas i s i n c r e a s i n g accounted f o r on the b a s i s retardation  Two flow  at a r a t e g r e a t e r than can be  o f i d e a l shock tube t h e o r y .  o f the contact s u r f a c e  The  i n t h i s p r e s s u r e range has  a l r e a d y been a t t r i b u t e d t o the r e f l e c t e d r a r e f a c t i o n wave. two  flow regimes w i l l be r e f e r r e d  t o as the "boundary  The  layer  l i m i t e d " regime (p^ < 4 T o r r ) and the " r a r e f a c t i o n wave l i m i t e d " regime (p, > 4 T o r r ) .  98 In f o r m u l a t i n g shock-heated gas, 1 be t r e a t e d f i r s t .  a model t o e x p l a i n the observed l e n g t h of  , the "boundary l a y e r l i m i t e d " regime w i l l We  s h a l l assume t h a t a l e n g t h  mixes w i t h the d r i v e r gas Since H o o k e r ^ ^  has  2  and  of t e s t  gas  t h e r e f o r e cannot c o n t r i b u t e to 1 .  shown t h a t the l e n g t h of the column of  mixed d r i v e r and t e s t gases i s independent of downstream p r e s s u r e , we  are j u s t i f i e d i n assuming t h a t  does not depend on p^.  a l s o assume t h a t of the a v a i l a b l e gas downstream of ^ , ( jPi)  fraction  x  r e t a i n e d , the remainder b e i n g  1 S  a v i s c o u s boundary l a y e r .  Conservation  only a lost  through  of mass then r e q u i r e s  t h a t the l e n g t h of the shock-heated gas behind the tube i s given  We  a shock at x  in  Q  by  x  A* 1  Since we  -ft / M  =  X  do not know the dependence o f  (6.8) ^  on x we  shall  evaluate  the i n t e g r a l approximately.„-We assume t h a t a l l the t e s t o r i g i n a l l y between  and some p o s i t i o n L /5,  heated l a y e r , so t h a t a constant  = 1 for  ^9(p,) *= 1  fraction  0  r e t a i n e d , the remainder being layer. o f gas  We  define  ^9(f>,)  •£ x i L> .  For L  2  < x 4  x , Q  l o s t through a v i s c o u s boundary (*o,f/) .  Since we  expect the l o s s separation  contact s u r f a c e ) ,  .fl(X,p,)  With these  /3 CP.) = A<X,P.)  I  f u r t h e r assume t h a t L  Pj«  2  i s assumed to be  to i n c r e a s e w i t h x (or e q u i v a l e n t l y , w i t h the  of shock and  We  remains i n the shock-  of the t e s t gas  (i,  =  0  2  gas  0  2  for  X * Xo  (6.9)  i s independent of downstream p r e s s u r e .  assumptions  (6.8)  becomes  = / 5 M * o - l 2 ) * JJCU-L.)  ( 6 > 1 0  ,  99 . f o r p-^ < 4 T o r r .  By d i f f e r e n t i a t i o n , ^  i s found  D  t o be  rl  1°  j> d t UtJ =; X  or  where the v e l o c i t i e s  are measured at x .  That i s , fi i s the  Q  Q  r a t i o o f the i d e a l d e n s i t y r a t i o t o the apparent d e n s i t y r a t i o determined from the r e l a t i v e v e l o c i t i e s surface at x . Q  From equation  of shock and contact  (6.5),  jfe  (6,12)  which can be e v a l u a t e d from Figure 6-8. Equation  (6.10) p r e d i c t s t h a t a graph o f  P lm  —J ~ 1  P should be a s t r a i g h t (L  2  - L ) and i n t e r c e p t ( x 1  l i n e f o r p^ < 4 T o r r , .with Q  - L ).  a  Po  slope  Figure 6-9 shows t h a t  2  against  this  i s indeed the case and g i v e s L, = 56 cm, L = 89 cm, f o r x = 120 cm 1 ' 2 o 0  Two c o n d i t i o n s under which the i n t e g r a l w r i t t e n as i n (6.10) f o r constant There the normalized  instantaneous  L  2  o f (6.8) can be  are shown i n Figure 6-10.  gas l o s s  i s p l o t t e d as a f u n c t i o n of d i s t a n c e x, where the f r a c t i o n a l gas loss  i s d e f i n e d as  SO^P.)  =  I-  /3 (X (  j P (  and  f Cp,) = • i - /S0Cp.> 0  )  (6.14)  0.5  40 i 0.4  1  0.6  r 0.8  1  1.0  ,  1  1  1.2  1.4  « 1.6  B  F i g u r e 6-9  E x p e r i m e n t a l Check on the V a l i d i t y of E q u a t i o n  (6.10)  ! 1.8  f 2.0  101  (a)  S( ,Pi) s e p a r a b l e f u n c t i o n of x and p^ o r a p p r o x i m a t e l y s o j t h a t i s , ^ ( X j P ^ ) changes l i t t l e f o r a l a r g e change i n p^. x  F i g u r e 6-10  l s  a  Two S i t u a t i o n s i n which the I n t e g r a l of E q u a t i o n (6.8) may be approximated as i n E q u a t i o n (6.10).  102 In  case  ( a ) , t h e more l i k e l y  s i t u a t i o n , G(x,p) e i t h e r i s  independent  o f p ^ o r depends v e r y w e a k l y on p ^ o v e r  range o f  < x < x ;  - -" G u , p . ) =  i  i*; \  g  e  the whole  n*)  =  ? ,  (6.15)  5 o ^ P.)  where 7^  depends  function  o f x and P j  From  o n l y on x.  (6.9)  J^Cx^f?)  Then  '  :  _,  .  '  ,  .  „  ?(X,P,) £ f (p,)  We  can r e w r i t e  fj From  7J(X) .6 I  Oi  t h a t  (6.8) i n t e r m s o f  C 6  1 7 )  (  (  ?(*>P,)  d  (6.18)  x  (6.16)  ^()C)dx L  Now  '  C ( * J P I ) as f o l l o w s :  = P CXo-L,) - J>,/  m  , ,  for * -  0  5 0  i s a separable  t h e i n t e g r a l must h a v e d i m e n s i o n s  define  such  0  (6.18) t h e n  ?S Upon s u b s t i t u t i n g identical  We  therefore  X  0  " L  (6.20)  z  (6.17)  x„ - L < x „ - L, ; o 2 o 1 Equation  of length.  =  0  from  .  that  J.X >9U)cU where,  (6.19)  i.e. L  0  1  ?O(P,)(XO-  f Cp,) = l-/3 (p,)  t o (6.10).  > L, l  becomes  J^^ " ') " f,  =  9  L  o  0  From  (6.20), L  2  LJ  , equation is clearly  ( 6 > 2 1 )  (6.21) becomes independent  of p ^  103. In case (b) GCx,p ) may 1  the curves  depend s t r o n g l y on p^,  provided  f o r v a r i o u s p^ d i f f e r only over small i n t e r v a l s  t o t a l l e n g t h A << x  - L^.  ^ ( X j p ^ ) w i l l then be separable and X ,  most o f the r e g i o n between approximately  of  ;  q  f o r I ^ independent of  and  over  (6.10) w i l l again h o l d  p^.  For p^ g r e a t e r than about 4 T o r r , the graph o f F i g u r e 6-9  becomes a h o r i z o n t a l l i n e .  We  seek an i n t e r p r e t a t i o n of  t h i s phenomenon i n terms of the r a r e f a c t i o n wave which  enters  the r e g i o n o f shock-heated gas  and a f f e c t s the flow i n t h i s  pressure  shows,  range.  As  Figure 6^8  approaches 4 T o r r .  We  |3  0  approaches u n i t y as  p^  s h a l l t h e r e f o r e assume that boundary  l a y e r s are n e g l i g i b l e f o r p^ g r e a t e r than 4 T o r r .  Now,  due  to  the presence of the r e f l e c t e d r a r e f a c t i o n wave i n the shockheated gas, we would expect the d e n s i t y o f t h i s r e g i o n to be non-uniform and to have an average value As b e f o r e , a constant removed by  length,  JOz.  , where  , of t e s t gas  i n i t i a l mixing p r o c e s s e s .  as before by  As /3  differ.  =  In t h i s " r a r e f a c t i o n l a y e r has  a  P,(*o-L.)  i s of the same form as  (6.22)  (6.10) and  ^  may  be  found  differentiating:  already pointed 2  1  by  4^ T h i s equation  >  i s assumed to be  wave l i m i t e d " regime, then, the shock-heated gas thickness, 1 , given  (b  out, however, the i n t e r p r e t a t i o n s of  Equation  n-1 independent of p 2  (6.2 2) s t a t e s t h a t  ^  should  ' ^  f o r p^ g r e a t e r than 4 T o r r and  fi  t  be  therefore  and  104 s u c c e s s f u l l y accounts; f o r the h o r i z o n t a l p o r t i o n of the i n F i g u r e 6-9.  Moreover, the i n t e r c e p t , x L  Q  graph  - L^, g i v e s  = 5.9 cm  1  (6.24)  i n good agreement w i t h the value of  = 56 cm o b t a i n e d  from  the "boundary l a y e r l i m i t e d " flow regime (see page 99). t h i s sense, then, the equations d e s c r i b i n g the two  In  flow regimes  are c o n s i s t e n t . Substituting =  1 t  (6.13) i n t o (6.10) and  0.259  3  o  + 0.275  (6.22) y i e l d s  , p, < 4 T o r r l (6.25)  h  x  -S1  =  0.508  B  , p  2  x  > 4 Torr  th  Using the experimental v a l u e s of 8 p l o t t e d i n F i g u r e 6-6. determined  The  and 8 , 7  these equations  are  agreement w i t h the e x p e r i m e n t a l l y  v a l u e s of 1 ^ 1 ^  i s seen to be v e r y good.  In s p i t e  of i t s crude nature the model t h e r e f o r e seems to e x p l a i n the observations. In summary, i t seems t h a t we t h i c k n e s s of shock-heated  can e x p l a i n the  gas by a simple p h y s i c a l model.  regimes of flow are apparent  l e n g t h , L^, of t e s t gas  with the d r i v e r gas. and a p o s i t i o n L , 2  Two  a c c o r d i n g as the downstream p r e s s u r e ,  p^, i s l e s s than or g r e a t e r than about 4 T o r r . a constant i n i t i a l  observed  In both  cases,  i s assumed to mix  For low p r e s s u r e s , t e s t gas between  independent  of downstream p r e s s u r e , i s  assumed to c o l l e c t without  l o s s between the shock f r o n t  contact surface.  0  Between L  and  and the o b s e r v a t i o n s t a t i o n at x , o a v i s c o u s boundary l a y e r then i s assumed to remove a constant L  105 f r a c t i o n of the gas encountered by the shock f r o n t . pressure  In the high  regime, the flow i s assumed t o be i n v i s c i d .  A f t e r the  i n i t i a l mixing over l e n g t h L^, the shock-heated l a y e r i s assumed t o r e t a i n a l l the t e s t gas e n t e r i n g i t . The o v e r t a k i n g o f the contact  s u r f a c e by the r e f l e c t e d r a r e f a c t i o n wave i s invoked to . e x p l a i n the l e n g t h and growth r a t e of t h i s l a y e r . I t should be emphasized t h a t when t h i s occurs  the p r o p e r t i e s o f the shock-  heated gas are a l t e r e d and the shock tube l o s e s i t s u s e f u l n e s s . The  model p r e d i c t s a l e n g t h o f shock-heated gas c o n s i s t e n t with  the observed value  for  i n the range 0.5 t o 15 T o r r .  d i f f e r e n t d r i v e r c o n d i t i o n s or o b s e r v a t i o n V"  c  = 10 KV and X  q  6.3  Currents  s t a t i o n s (we used  = 120 cm) , the c r i t i c a l p r e s s u r e  the two flow regimes  For  separating  (4 T o r r i n our case) i s expected to change.  i n the Shock-Heated Gas  In some e l e c t r o m a g n e t i c a l l y d r i v e n shock tubes, c u r r e n t s have been d e t e c t e d  discharge  i n the shocked gas immediately  f3 4*) behind  the shock f r o n t  v  * . J  Such c u r r e n t s a l t e r the c o n d i t i o n s  o f the shock-heated gas and render standard  shock wave theory.  i n v a l i d the a p p l i c a t i o n of  The experiments of Chapter 4 i n d i -  cate t h a t the behaviour o f shock waves produced i n the present shock tube i s decoupled from the a c t i v e stage  o f the e l e c t r i c a l  discharge  s i n c e the diaphragm does not begin  discharge  c u r r e n t has f a l l e n to a small f r a c t i o n of i t s peak  value.  (See e.g. F i g u r e 4-11.  For V  c  to break u n t i l the  = 10 KV and diaphragms  0.005 i n c h t h i c k , the diaphragm breaks when the c u r r e n t has f a l l e n t o about 10% o f i t s peak value.)  A l s o , a f t e r the  \  106 diaphragm does break, the c u r r e n t decays to an  undetectably  3 s m a l l value expedient,  (< 10  amps) w i t h i n a few microseconds.  I t was  however, to examine the shock-heated gas  felt  f o r small  currents. >:  For t h i s purpose, a s m a l l 100-turn c o i l  ;  2.5  mm  i n diameter was  p o s i t i o n 136  cm  differentially t o g e t h e r with the c o i l with axes.  and  The  s i g n a l was  amplified  d i s p l a y e d on a dual-beam o s c i l l o s c o p e  the s i g n a l from a pressure probe l o c a t e d  at the same a x i a l p o s i t i o n .  the c o i l  long  i n s e r t e d i n t o the shock tube at a  from the diaphragm. and  5 mm  Observations  axis a l i g n e d along three mutually  R e s u l t s are shown i n Figure 6-11.  Due  opposite  were made  perpendicular  to the a c o u s t i c  delay i n the pressure probe, i t s s i g n a l i s delayed  18 micro-  seconds with r e s p e c t to the shock a r r i v a l . The  coil  s i g n a l s appear roughly  o r i e n t a t i o n s of the c o i l . may  be  T h i s f a c t suggests t h a t the s i g n a l  caused by c a p a c i t a t i v e c o u p l i n g between the  heated plasma and the c o i l . s i g n a l disappears  w i t h i n 30 microseconds i n a l l cases  the d i s c h a r g e  d r i v e r gas  as w e l l .  the c o i l  to a c u r r e n t  However, a d m i t t i n g  associated i n the  such a c u r r e n t  i n the d r i v e r gas  the  strongly  s i n c e such a c u r r e n t would p e r s i s t  assuming t h a t i t goes, undetected reason,  shock-  Furthermore, the f a c t t h a t  i n d i c a t e s t h a t the s i g n a l i s not due with  the same f o r a l l  and  f o r some  s i g n a l amplitude p l a c e s an upper l i m i t  of  about 2 amps on the c u r r e n t .  Since the plasma c o n d u c t i v i t y i s f3 3 1 r e l a t i v e l y h i g h under the c o n d i t i o n s used ( W i l l i a m s o n 3 -1 estimates i t to be about 4'10 mho-m ) , h e a t i n g produced by v  v  Shock Tube  Ceil Horizontal P a r a l l e l to Tube A x i s  Probe Coil  Coil Vertical Perpendicular to Tube A x i s  Coil Horizontal Perpendicular to Tube A x i s F i g u r e 6-11  Attempts t o Observe C u r r e n t s i n Shock-Heated Gas B e h i n d Mach 16 Shock i n Argon a t 1 T o r r Upper Beam: P r e s s u r e Probe 0.2 V / d i v i s i o n Lower Beam: Search C o i l 0.5 V / d i v i s i o n Time ( R i g h t t o L e f t ) : 20 u s e c / d i v i s i o n  108 ohmic d i s s i p a t i o n o f t h i s c u r r e n t i s much s m a l l e r than the energy added by shock wave h e a t i n g . is  i n fact a current  The c u r r e n t  - i f there  i n the shock-heated gas - w i l l then have  n e g l i g i b l e e f f e c t on the p r o p e r t i e s o f the shock-heated gas.  6.4  P r o p e r t i e s o f the Shock-Heated Gas In s e c t i o n 6.2, the measured t h i c k n e s s  of shock-heated  gas was shown t o i n c r e a s e a t l e s s than the p r e d i c t e d r a t e f o r pressures across  below about 4 T o r r .  The apparent d e n s i t y  ratio  the shock f r o n t was t h e r e f o r e g r e a t e r than expected.  I f these e f f e c t s are a t t r i b u t a b l e t o leakage o f gas past the contact still gas  s u r f a c e v i a a v i s c o u s boundary l a y e r , then one would  expect the thermodynamic p r o p e r t i e s of the shock-heated  t o be the p r e d i c t e d v a l u e s .  In t h i s s e c t i o n these p r o p e r t i e s  are i n v e s t i g a t e d . (a)  Pressure  Behind the Shock Front  - From equations strong  shock wave ( p  2  (2.1) and (2.5) i t f o l l o w s t h a t f o r a >> p-^, and u^ >> u ) the p r e s s u r e , p , 2  2  behind the shock i s given t o a good approximation by P where  ftVg  (6.26)  i s the t e s t gas mass d e n s i t y and V  i s the shock speed  i n the l a b o r a t o r y .  V  g  X  2  and  p^  g  can be measured and a s i g n a l of  amplitude A, p r o p o r t i o n a l t o the shock-heated gas p r e s s u r e , can be obtained  from a pressure  to the t e s t gas p r e s s u r e , A  probe.  Since  p , 2  i s proportional  p^, (6.26) p r e d i c t s t h a t  oc p.V?  (6.27)  109 The  measured  in  Figure  is  seen t h a t  a m p l i t u d e , A,  6-12  o f a p r e s s u r e probe  and  response  the probe Since  •  s  A i s proportional quantity,  is plotted  2 o f t h e m e a s u r e d v a l u e o f P-jV « 2  as a f u n c t i o n  of the l a t t e r  signal  to P ^ ,  except f o r large values  v  g  f o r which  large values of p  are  2  expected  i s probably non-linear.  the p r e s s u r e , p ,  has  2  the expected  linear  dependence  2 on P j V ,  i t seems v e r y l i k e l y  g  of Rankine-Hugoniot  theory.  that p One  p r e s s u r e measurement t o p r o v e technique  for calibrating  conforms  2  t o the  s h o u l d attempt  this  predictions  an a b s o l u t e  c o n c l u s i v e l y , b u t an a c c u r a t e  t h e p r e s s u r e p r o b e , s u c h as  that  (391 d e v e l o p e d by K a t s a r o s (b)  Sound Speed The  temperature sound  , was  not  available,  i n the Shock-Heated  purpose  of t h i s  e x p e r i m e n t was  of the shock-heated  i n the gas.  The  Gas  gas by  measured  pared w i t h the v a l u e p r e d i c t e d  t o measure  o b s e r v i n g the speed  temperature by  the  could  t h e Augmented  t h e o r y o f C h a p t e r 2. sound speed, a , i s g i v e n a p p r o x i m a t e l y by For. a s h o c k - h e a t e d gas A h l b o r n ^ ^ has  t h e n be  of com-  Rankine-Hugoniot  2  <3  2  Here g  i s the e f f e c t i v e  2  =  • k 23  adiabatic  shown t h a t  the  (6.28)  I±  exponent  of the  shock-heated  gas g i v e n by  where u  2  and  h  2  h  2  u  2  are the s p e c i f i c  enthalpy, respectively.  Graphs  internal of g  2  energy  and  as a f u n c t i o n  specific of  (351 temperature  a r e given, by A h l b o r n and S a l v a t  v  1  .  The p r e s s u r e ,  120 ©  F i g u r e 6-12  Dependence of P r e s s u r e Probe S i g n a l , A m p l i t u d e on P-.V2  p ,  c a n be  2  assuming  expressed  the  shock-heated  temperature, is  the  density  as  T .  e\  Here  2  heavy p a r t i c l e i s given  gas  i s the  from  i s the  free  i n thermal  degree  number d e n s i t y  e q u i l i b r i u m at  of i o n i z a t i o n  (atoms and  and  ions).  n  The  2  mass  by  fx where m  t o be  mass o f  =  an  e l e c t r o n s has  "2™  (6.30)  atom o r i o n .  The  been ignored.  c o n t r i b u t i o n to  Equation  (6.23)  p  then  becomes a* If  ionization  $  ^  is negligible  monatomic g a s , heats,  =  g  , and  (low  becomes t h e  2  we  get  —  —  —  (6.3i)  temperatures),  familiar  ratio  the u s u a l e x p r e s s i o n  then,  of  for a  specific  f o r the  sound  speed a  Even  and  good e s t i m a t e decrease  increases.  ~m—  =  f o r a s m a l l degree of  fairly C<  *  ( 6 , 3 2 )  ionization  equation  (6.32) g i v e s  o f . t h e sound speed s i n c e the  of g  2  Hence f r o m  compensate e a c h o t h e r a measurement o f a , 2  increase  somewhat as T  2  can  be  T  a of  2  obtained  approximately. To m e a s u r e a , 2  the with  shock-heated  gas  a time-of-flight  detect  the p u l s e .  c a p a c i t o r was  a pressure  and  the  speed  technique  Electrical  discharged  d i s t u r b a n c e was of t h i s  using  energy  through  p u l s e was  a pressure f r o m a 0.25  a tiny  produced i n  spark  gap  measured  probe  to  yfarad (Figure  6-13)  Triggerable A i r Spark Gap S w i t c h  Tungsten Wire diameter  Insulated Copper Strips Clamped Together  Set Screw  Capacitor 0.25  2  iifarad  mm. to  C i r c u i t Parameters: R i n g i n g P e r i o d = 3 |isec 5 L o g a r i t h m i c Decrement = 1.7*10  F i g u r e 6-13  _ i sec  D e t a i l s o f Spark Gap Used t o Produce "Sound" P u l s e  113 on one w a l l of the shock tube to produce a p-ressure p u l s e the shock wave had passed  t h a t p o i n t i n the shock tube.  p r e s s u r e probe i n s e r t e d through  (See F i g u r e A-3 i n the  Appendix f o r t y p i c a l p r e s s u r e probe s i g n a l s . ) through  A  the opposite s i d e o f the shock  tube d e t e c t e d the a r r i v a l o f the p u l s e .  propagated  after  Since the p u l s e  f l o w i n g gas, i t was necessary  to displace  the d e t e c t i n g probe a x i a l l y downstream i n order t o observe the p u l s e , as shown i n F i g u r e 6-14.  On subsequent shots under  i d e n t i c a l • c o n d i t i o n s the a r r i v a l time o f the p u l s e was measured f o r d i f f e r e n t d e t e c t o r probe l o c a t i o n s i n order t o determine the sound speed. The  p r e s s u r e p u l s e amplitude  w i t h d i s t a n c e from the source)  was f i n i t e  (and decreased  so one would expect  the speed of  p r o p a g a t i o n t o be g r e a t e r than the true sound speed.  In order  to minimize t h i s e f f e c t , the c a p a c i t o r charging v o l t a g e was reduced u n t i l the p u l s e was j u s t d e t e c t a b l e f o r a t y p i c a l p o s i t i o n o f the probe. 4.4«10  A s u i t a b l e v o l t a g e was found  t o be  volts.  3  To determine the r e l a t i o n between the p u l s e  propagation  speed and the a r r i v a l time at the probe, c o n s i d e r the s i t u a t i o n of F i g u r e 6-14. hemispherically.  The a c o u s t i c source  i s assumed to r a d i a t e  The probe then d e t e c t s t h a t p o r t i o n of the  p r e s s u r e d i s t u r b a n c e which has a net v e l o c i t y g i v e n by V, = s  where v  2  V  £  + a  (6.33)  2  i s the gas flow v e l o c i t y  p u l s e w i t h r e s p e c t to the gas.  and a  In the  2  i s the v e l o c i t y o f the two-dimensional  Shock Front  S m a l l Spark Gap'  Net V e l o c i t y - o f "Sound" P u l s e  J  S l i d i n g P r e s s u r e Probe Mount  A  v  2  Gas Flow  -Pressure Probe  F i g u r e 6-14 Arrangement f o r M e a s u r i n g Sound Speed i n Shock-Heated Gas  115 c o - o r d i n a t e system o f Figure 6-14 (i and * are u n i t v e c t o r s ) , Vs,  j - i + -|- J  =  (6.34)  where t i s the source-to-probe time o f f l i g h t Substituting  (6.34) and (6.35) i n t o  -  ^  (6.33)  (t* -  s  o f the p u l s e .  + T" ^  -36:>  (6  Thus  aJtV  =  ( S -  V t)  +  z  A  2 ,  (6.37)  The p u l s e speed, a , can then be found by measuring the a r r i v a l 2  2 time, t , f o r d i f f e r e n t s and A and p l o t t i n g  (s - v t ) 2  2 +A  2 against t .  The gas flow speed, v , i s c a l c u l a t e d 2  from the  Augmented Rankine-Hugoniot e q u a t i o n s . T h i s procedure was c a r r i e d out f o r a Mach 13 shock i n argon a t p^ = 1 T o r r and t h e — r e s u l t s are shown i n Figure 6-15. The f i n a l p a r t o f the graph i s seen t o be a good s t r a i g h t From the slope we f i n d a  2  = 2140 m/sec.  line.  However, as a l r e a d y  e x p l a i n e d , t h i s value i s g r e a t e r than the true sound speed, and a c o r r e c t i o n was obtained as f o l l o w s .  The amplitude o f the spark-  produced d i s t u r b a n c e , &p, was observed to be about 201 o f the p r e s s u r e behind the main shock, p . 2  (See e.g. Figure  A-3(c)(ii)).  The Mach number of the d i s t u r b a n c e i s then found from (2.14) t o  M=  be  where P / p 2  2  [ I F  = 1  +  {<* >fyfc-U-i)}] +l  Ap/p  2  Y2  = 1.2 i 5%.  = Lost  3%  D i v i d i n g the measured  d i s t u r b a n c e speed by i t s Mach number y i e l d s a b e t t e r estimate of the sound speed, a  £  = 1980 m/sec £ 5%. Using (6.32)  the gas temperature as T  ?  = llll-10  4  °K S 10%.  then gives  116  100  200  2  t F i g u r e 6-15  2  300  (jisec )  D e t e r m i n a t i o n o f Sound Speed i n Shocked Argon  117 T h i s measured temperature agrees w e l l w i t h t h e v a l u e o f 1.08'10^°K p r e d i c t e d by the Augmented Rankine-Hugoniot e q u a t i o n s f o r a Mach 13 shock i n argon a t 1 T o r r .  The a p p r o x i m a t i o n  made i n r e p l a c i n g (6.31) by (6.32) i s q u i t e good s i n c e a t t h i s temperature  0\, = 0.07 and g  2  i s found i n r e f e r e n c e 35 t o be  about 1.45. Thus, i n (6.32) a v a l u e o f & = 1.55 s h o u l d have been used i n s t e a d o f  $ = 1.67.  Furthermore, A h l b o r n ^ ^ places  a l i m i t o f about 121 on t h e e r r o r o f (6.28) from which  (6.31)  was d e r i v e d , so t h e use o f (6.32) i s j u s t i f i e d i n t h i s c a s e . W i t h i n the a c c u r a c y o f o u r c a l c u l a t i o n s , t h e n , i t appears t h a t the shock-heated gas temperature conforms t o theoretical predictions. the  This f a c t , together w i t h the study of  shock-heated gas p r e s s u r e , i s good e v i d e n c e t h a t t h e shock-  h e a t e d gas i s i n a s t a t e o f e q u i l i b r i u m determined by t h e Augmented Rankine-Hugoniot e q u a t i o n s o f Chapter 2.  T h i s con-  c l u s i o n a p p l i e s , o f c o u r s e , only i n t h e r e g i o n o f p r e s s u r e and shock speed s t u d i e d ( i . e . p^ 52/ 1 T o r r , M W 1 3 ) .  No experiments  were performed i n the " r a r e f a c t i o n wave l i m i t e d " f l o w regime, (c)  U n i f o r m i t y o f t h e Shock-Heated Gas The u n i f o r m i t y o f the r a d i a t i o n e m i t t e d by t h e shock-  heated gas, as seen i n t h e smear photographs  (Figure 6-3),  suggests t h a t t h i s r e g i o n i s one o f f a i r l y u n i f o r m properties.  thermodynamic  S i n c e the l u m i n o s i t y i s a s e n s i t i v e f u n c t i o n o f  t e m p e r a t u r e , any temperature v a r i a t i o n i n t h i s r e g i o n s h o u l d produce a n o t i c e a b l e n o n - u n i f o r m i t y i n the luminous gas. I t was shown i n Chapter 5 ( F i g u r e 5-1) t h a t t h e shock a t t e n u a t i o n i s low (about 20% p e r meter f o r p, < 5 T o r r ) .  118 . Low  a t t e n u a t i o n i s a r e q u i r e m e n t f o r the shock-heated  f a i r l y u n i f o r m i n the a x i a l d i r e c t i o n , i s not c o n s t a n t , s u c c e s s i v e have d i f f e r e n t  gas t o be  f o r i f the shock  speed  a x i a l samples i n the heated gas  will  p r o p e r t i e s s i n c e each w i l l have been heated by a  shock wave of d i f f e r e n t  strength.  F i n a l l y , the s u c c e s s of the sound speed measurements suggests t h a t the hot gas p r o p e r t i e s are u n i f o r m i n b o t h and a x i a l d i r e c t i o n s and r e p r o d u c i b l e from shot t o s h o t .  radial  119 CHAPTER 7  1.1  CONCLUSIONS  Conclusions In Chapter 1, a number o f q u e s t i o n s r e g a r d i n g the  o p e r a t i o n o f the Smy  shock tube were posed.  Answers  t o these  q u e s t i o n s have been o b t a i n e d . I t has been demonstrated t h a t the shock tube behaves i n a manner s i m i l a r t o c o l d - d r i v e r diaphragm shock t u b e s . magnetic f i e l d produced by the " b a c k s t r a p " has  The  negligible  i n f l u e n c e on the a c c e l e r a t i o n of the d r i v e r gas.  The h i g h shock  speeds are a t t r i b u t a b l e t o the h i g h sound speed and p r e s s u r e b r o u g h t about i n the h e l i u m d r i v e r gas by the e l e c t r i c a l discharge. We have developed a t e c h n i q u e t o determine the heated d r i v e r gas p r e s s u r e , p^.  From a measurement of the time  r e q u i r e d f o r the diaphragm t o open c o m p l e t e l y , t  , p^ can be  c a l c u l a t e d u s i n g a s i m p l e , e x p e r i m e n t a l l y - v e r i f i e d model. find  We  that p  f o r 5 KV < V  4  = 0.332 V  < 15 KV, where V  i n k i l o v o l t s and p  4  (7.1)  2  i s t h e c a p a c i t o r bank v o l t a g e  i s i n atmospheres.  A l t h o u g h o t h e r shock  tubes of a s i m i l a r type have been d e v e l o p e d ^  2  ^ , we have  been u n a b l e t o l o c a t e i n the s t a n d a r d l i t e r a t u r e any r e f e r e n c e to u s e f u l measurements o f p . 4  Of the e l e c t r i c a l energy s t o r e d i n the c a p a c i t o r bank, about 50% i s d i s s i p a t e d i n the d i s c h a r g e p l a s m a , and about h a l f of t h i s energy i s e f f e c t i v e i n r a i s i n g the d r i v e r gas p r e s s u r e .  .  120  I  i  We have found t h a t s t a n d a r d o n e - d i m e n s i o n a l shock tube t h e o r y cannot be a p p l i e d d i r e c t l y t o t h i s shock tube f o r the purpose o f d e t e r m i n i n g the shock speed from the diaphragm p r e s s u r e r a t i o , p^/p^.  I n s t e a d , we f i n d t h a t the shocks behave  as i f d r i v e n by an e f f e c t i v e d r i v e r p r e s s u r e , p^ , l e s s than the a c t u a l d r i v e r p r e s s u r e , p , measured from the diaphragm opening 4  time.  E m p i r i c a l l y , we f i n d  that  P4 = C P ) ' 0  (7.2)  6 3  4  for  15 atm. < p  4  < 70 atm., where p  4  and p^ are e x p r e s s e d i n  atmospheres, and t h a t  where a^ i s the e f f e c t i v e sound speed o f the d r i v e r gas, and is  J>^  the d r i v e r gas mass d e n s i t y which can be c a l c u l a t e d from the  initial  helium f i l l i n g  c o n s t a n t volume.  We  p r e s s u r e s i n c e the gas i s heated a t  a t t r i b u t e the d i s c r e p a n c y between a c t u a l  and e f f e c t i v e d r i v e r p r e s s u r e s t o the t h r e e - d i m e n s i o n a l motion of  the d r i v e r gas as i t passes through the opening diaphragm. U s i n g the r e l a t i o n s  (7.1) t o (7.3) and s t a n d a r d shock  tube t h e o r y , one can then p r e d i c t the shock speed from the f r e e p a r a m e t e r s , the t e s t and d r i v e r gas f i l l i n g p r e s s u r e s , and the discharge voltage V . In  o r d e r to determine the l e n g t h of the shock-heated gas  s l u g c o n t a i n e d between the shock f r o n t and the c o n t a c t s u r f a c e , we have used a shock r e f l e c t i o n t e c h n i q u e . T h i s t e c h n i q u e i d e n t i f i e s not o n l y the shock f r o n t i n smear camera photographs of  the gas f l o w , but a l s o t h e " c o n t a c t s u r f a c e s i n c e the  121 r e f l e c t e d shock i s r e f r a c t e d on p a s s i n g  through t h i s s u r f a c e .  We  have measured t h e l e n g t h o f the shock-heated gas s l u g , 1 , a t a p o s i t i o n i n the shock tube 120 cm from the diaphragm  fora  c a p a c i t o r v o l t a g e o f 10 KV and f o r argon t e s t gas p r e s s u r e s , p^, i n the range 0.5 T o r r < p^ < 15 T o r r .  In a l l cases we have  found t h a t 1  t n  i s l e s s than t h e v a l u e , l ,  shock tube t h e o r y . 1  m  p r e d i c t e d by i d e a l  F u r t h e r m o r e , we have found from our s t u d y o f  and i t s r a t e o f i n c r e a s e t h a t under t h e above c o n d i t i o n s two  regimes o f gas f l o w can be d i s t i n g u i s h e d by whether p^ i s l e s s than o r g r e a t e r than 4 T o r r . to  e x p l a i n our r e s u l t s .  initially  A simple model has been  formulated  For a l l p^, we f i n d t h a t t e s t gas  i n the f i r s t 60 cm o f the shock tube mixes w i t h the  d r i v e r gas and so i s l o s t .  For  < 4 T o r r , where d± /dt < m  d l ^ / d t , a f u r t h e r l o s s mechanism, p r o b a b l y a v i s c o u s l a y e r , removes some o f the shock-heated gas.  boundary  For p^ > 4 T o r r ,  where dl^/dt....> d l ^ / d t , the gas f l o w appears t o be dominated by the r a r e f a c t i o n wave r e f l e c t e d from the d r i v e r s e c t i o n . wave o v e r t a k e s and r e t a r d s the c o n t a c t s u r f a c e .  This  Whereas the  removal o f shock-heated gas does n o t a l t e r .the p r o p e r t i e s o f the r e t a i n e d g a s , the presence o f the r a r e f a c t i o n wave does a l t e r the s h o c k - h e a t e d gas p r o p e r t i e s and so cannot be t o l e r a t e d . knowledge  A  o f t h e l i m i t a t i o n s o f a shock tube i n t h i s r e s p e c t i s  essential. Our i n v e s t i g a t i o n s show t h a t the l e n g t h o f t h e shockheated gas sample, under the above c o n d i t i o n s , v a r i e s from 5 cm for  a Mach 20 shock i n argon a t 0.5 T o r r t o 12 cm f o r a Mach 12  shock i n argon a t 3 T o r r , a t a p o s i t i o n 120 cm from the diaphragm.  122 The p r e s s u r e , p , b e h i n d the shock f r o n t was s t u d i e d 2  w i t h a p i e z o e l e c t r i c p r e s s u r e probe and was found t o agree w i t h the p r e d i c t i o n s o f Rankine-Hugoniot t h e o r y .  A l s o , the tempera-  t u r e , T , of the shock-heated gas was d e t e r m i n e d from the sound 2  speed, a , which was measured by a p p l y i n g a t i m e - o f - f l i g h t 2  t e c h n i q u e t o a s m a l l p r e s s u r e d i s t u r b a n c e i n t r o d u c e d i n t o the hot gas.  Behind a Mach 13 shock i n a r g o n . i n i t i a l l y a p^ = 1 T o r r  ( i n which case the degree o f i o n i z a t i o n i s c a l c u l a t e d t o be 7 1 ) , the temperature measured i n t h i s way agreed v e r y w e l l w i t h the t e m p e r a t u r e c a l c u l a t e d from the shock speed u s i n g the t h e o r y o f Chapter 2.  I t a p p e a r s , t h e n , from t h i s study o f p  2  and T , 2  that  the shock-heated gas p r o p e r t i e s conform w e l l t o the e q u i l i b r i u m p r e d i c t i o n s o f the Rankine-Hugoniot t h e o r y i n the regime  studied.  In summary, our i n v e s t i g a t i o n s show t h a t t h e Smy shock tube i s an e x c e l l e n t s o u r c e o f plasma o f p r e d i c t a b l e  7. 2  properties.  S u g g e s t i o n s f o r F u t u r e Work Our i n v e s t i g a t i o n s o f the s e p a r a t i o n o f shock f r o n t and  c o n t a c t s u r f a c e showed t h a t the r a r e f a c t i o n wave r e f l e c t e d from the d r i v e r s e c t i o n dominated  the gas f l o w f o r p^ > 4 T o r r , a t a  bank v o l t a g e o f 10 KV and a t a p o s i t i o n 120 cm from the diaphragm. To e x t e n d the u s e f u l p r e s s u r e range o f the shock tube above 4 T o r r , an attempt s h o u l d be made t o l e n g t h e n the d r i v e r (Our d r i v e r s e c t i o n was 7.6 cm long.)  section.  T h i s would a l s o enable a  l a r g e r sample o f shock-heated gas t o be o b t a i n e d by moving the o b s e r v a t i o n s t a t i o n f a r t h e r away from the diaphragm.  123 In o r d e r t o make f u l l use o f t h e cap-acitor bank energy, an attempt t o e l i m i n a t e the e x t e r n a l spark gap s w i t c h be made. off  The d r i v e r s e c t i o n would have t o be m o d i f i e d  t h e d e s i r e d v o l t a g e s and t o i n c l u d e a " t r i g g e r "  should to hold  facility.  I t may be p o s s i b l e t o reduce energy l o s s e s from t h e h o t d r i v e r gas by u s i n g e l e c t r o d e m a t e r i a l s o f low t h e r m a l con-. ductivity  (e.g. s t a i n l e s s s t e e l ) and by l i n i n g the d r i v e r s e c t i o n (27  w i t h a heat r e s i s t a n t m a t e r i a l (e.g. Lexan^ e a s i l y evaporated.  281  '  J  ) which i s n o t  A l s o , d i l u t i o n o f the d r i v e r gas w i t h  high  m o l e c u l a r w e i g h t m a t e r i a l s e v a p o r a t e d from w a l l s and e l e c t r o d e s . i s u n d e s i r a b l e because t h i s reduces t h e d r i v e r gas sound speed. I n the p r e s e n t  shock tube d e s i g n , t h e h e a t e d d r i v e r gas  can c o o l a p p r e c i a b l y d u r i n g the i n t e r v a l the d i s c h a r g e  and t h e diaphragm  rupture.  ( »v 70 ysec) bet\\?een To overcome t h i s  p r o b l e m , and so i n c r e a s e t h e speed of shocks a v a i l a b l e , two capacitor discharges  c o u l d be u s e d , t h e f i r s t  t o open the  diaphragm, and the second t o heat the d r i v e r gas j u s t as the diaphragm b e g i n s t o open.  The second c a p a c i t o r bank s h o u l d  d i s c h a r g e q u i c k l y so t h a t the d r i v e r gas i s heated b e f o r e i t escapes.  124  BIBLIOGRAPHY 1.  A. C. K o l b , P h y s i c a l R e v i e w 1 0 7 , 345 (1957  2.  A. G. Gaydon and I . R. H u r l e , The Shock Tube i n H i g h Temperature C h e m i c a l P h y s i c s , CTTapman and H a l l , London  3.  H. Muntenbruch, p r i v a t e communication  4.  H. Muntenbruch, IPP 3/58 G a r c h i n g b e i Munchen  5.  P. R. Smy, Nature 193, 969 (1962)  6.  P. R. Smy,  7.  J . N. B r a d l e y , Shock Waves i n C h e m i s t r y and P h y s i c s , Methuen, London (1962)  8.  Ya. B. Z e l ' d o v i c h and Yu. P. 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C l o t f e l t e r , P h y s i c a l Review 82, 879 (1951)  (1966)  (1964)  (1963)  (1966)  127  APPENDIX - PRESSURE PROBES  The p r e s s u r e probes used i n the i n v e s t i g a t i o n s r e p o r t e d f40 411 i n t h i s t h e s i s are o f t h e b a r type developed by Edwards^ and S t e r n and D a c u s ^ ^ .  '  '  Many subsequent improvements and  m o d i f i c a t i o n s t o t h i s b a s i c probe d e s i g n and t o t h e c a l i b r a t i o n t e c h n i q u e s have been r e p o r t e d ^ * ^ ' ^ 5 0 ) ^  A n  excellent  of t h e p r e s s u r e b a r t e c h n i q u e has been g i v e n by Measures  review  J o n e s .  has i n v e s t i g a t e d the p r o p e r t i e s o f these probes  in great d e t a i l . The b a s i c d e s i g n i s i l l u s t r a t e d i n F i g u r e A - l .  The  p r e s s u r e t o be measured i s a p p l i e d n o r m a l l y t o the f r e e end o f the f r o n t r o d , A.  A s t r e s s wave propagates  along t h i s r o d t o a  p i e z o e l e c t r i c element B and then i n t o the r e a r r o d , C. E l e c t r i c a l c o n n e c t i o n s are made t o B i n a s u i t a b l e manner and a s i g n a l , which  i s p r o p o r t i o n a l t o the a p p l i e d f o r c e , can be  c o n v e n i e n t l y observed  on an o s c i l l o s c o p e .  T h i s type o f gauge has a number o f f e a t u r e s which make i t u s e f u l i n plasma p h y s i c s e x p e r i m e n t s .  The s e n s i t i v e element  Applied Pressure F r o n t Rod  Piezoelectric Element  Rear Rod  F i g u r e A - l B a s i c F e a t u r e s o f P i e z o e l e c t r i c P r e s s u r e Probe  128  and  the a s s o c i a t e d e l e c t r i c a l c o n n e c t i o n s  the r e g i o n o c c u p i e d  by the plasma.  the element o f t h e r m a l l y - i n d u c e d  can be removed out  I n t h i s way,  the e f f e c t  s t r e s s e s can be a v o i d e d .  the c i r c u i t r y can then be c o n v e n i e n t l y s h i e l d e d a g a i n s t magnetic f i e l d s a s s o c i a t e d w i t h d i s c h a r g e - p r o d u c e d cases where i t i s d i f f i c u l t t o exclude the measurement of the p r e s s u r e  Also,  electro-  The  In  completely,  acoustically  u n t i l a f t e r the r a d i a t e d f i e l d s have decayed by c h o o s i n g s u i t a b l e l e n g t h f o r r o d A.  on  plasmas.  radiated fields  can be d e l a y e d  of  a  delay i s  t ei-V lc  ( A  d  *  1 }  where c^ i s the s t r e s s wave speed i n the f r o n t rod o f l e n g t h 1 ^ . Good s p a t i a l r e s o l u t i o n can be o b t a i n e d , w i t h a s a c r i f i c e i n s e n s i t i v i t y , by u s i n g a f r o n t rod of s u f f i c i e n t l y s m a l l The  diameter.  o b s e r v a t i o n time depends on the l e n g t h o f the r e a r r o d .  The  s t r e s s wave, a f t e r p a s s i n g through the p i e z o e l e c t r i c element, e n t e r s the r e a r r o d . r e t u r n s and r e a r rod has  On r e f l e c t i o n from the f r e e end, i t  a g a i n a c t i v a t e s the p i e z o e l e c t r i c element. l e n g t h 1 ^ and a s t r e s s wave p r o p a g a t i o n  Cj, the o b s e r v a t i o n  I f the  speed of  time i s t  o  b  s  2 I 3 / C 3  (A.2)  and  can be of any d e s i r e d d u r a t i o n by s u i t a b l e c h o i c e o f 1 ^ .  The  r e a r rod s e r v e s  a n o t h e r purpose i n m e c h a n i c a l l y  l o a d i n g the  s e n s i t i v e element so t h a t o s c i l l a t i o n s o f the element at i t s n a t u r a l f r e q u e n c i e s are s u p p r e s s e d . In c o n s t r u c t i n g such a p r c b e , some c o n s i d e r a t i o n s must be borne i n mind.  The  t r a n s m i s s i o n of the s t r e s s wave between  the s e c t i o n s of the composite probe of F i g u r e A - l depends on  the  129 r e l a t i v e a c o u s t i c impedances of the adjacent" s e c t i o n s . a c o u s t i c impedance i s d e f i n e d  The  as  Z = fc j> = mass d e n s i t y  where  c = propagation I f Zj  (A. 3)  = Z  2  across  speed  \  the i n t e r f a c e between media 1 and  t r a n s m i s s i o n o f the s t r e s s wave w i l l must be  result.  2,  100%  Considerable  taken i n g l u e i n g the s e c t i o n s o f such a probe  care. .  together.  Even i f the s e c t i o n s themselves are a c o u s t i c a l l y matched, the bonding agent may pressure  cause r e f l e c t i o n s which c o u l d d i s t o r t  s i g n a l under o b s e r v a t i o n .  Jones^^  has  shown t h a t  the glue need not be a c o u s t i c a l l y matched to the r e s t of probe p r o v i d e d  the  the  the t h i c k n e s s , d, of the l a y e r o f g l u e i s s m a l l  compared t o the probe r a d i u s , r .  For a t y p i c a l epoxy r e s i n ,  he shows t h a t one must have d < 0.06r for  99%  transmission.  Care must be t a k e n to ensure t h a t  there  are no a i r bubbles i n the g l u e , however, s i n c e even e x t r e m e l y t h i n l a y e r s of a i r can cause severe r e f l e c t i o n s and of the  distortion  s i g n a l . In d e s i g n i n g  a p r o b e , one must a l s o c o n s i d e r  f r e q u e n c y spectrum of the p r e s s u r e On p r o p a g a t i n g dispersed;  the  v a r i a t i o n s t o be measured.  through the probe f r o n t r o d , p r e s s u r e  pulses  are  t h a t i s , d i f f e r e n t f r e q u e n c y components propagate  w i t h d i f f e r e n t speeds.  The  s o l u t i o n o f the a p p r o p r i a t e  elastic  equations  f o r the d i s p e r s i o n r e l a t i o n s i s a h i g h l y complex  problem.  However, i t can be shown t o a f a i r degree of accuracy  130 that i f X  where * ^ m  n  tn,n  ( A  i s the minimum wavelength p r e s e n t  *  4 )  i n the p r e s s u r e  s p e c t r u m , then the p u l s e w i l l propagate u n d i s t o r t e d w i t h speed c  o  = (E/p)  1 / 2  where E i s Young's modulus f o r t h e m a t e r i a l of the r o d . (See J o n e s f o r small radius.  references.)  One s h o u l d t h e r e f o r e use a r o d of  A more d e t a i l e d a n a l y s i s s h o w s  t h a t the  s i g n a l r i s e time o f a probe s u b j e c t e d t o a p r e s s u r e s t e p i s g i v e n by  . t-  where V  M  = Poisson's  (/  »*(•$-)'  (ir)  •  i.»r  ( A  .  5 )  r a t i o f o r the f r o n t rod  1^ = l e n g t h o f the f r o n t rod (The r i s e time i s d e f i n e d as t h e time r e q u i r e d f o r t h e s i g n a l t o r i s e from 10% t o 90% o f i t s f i n a l v a l u e . ) t h e r e f o r e be o b t a i n e d f o r s m a l l  Fast r i s e - t i m e s can  V , s m a l l r and l a r g e C .  The  q  r i s e - t i m e cannot, however, be l e s s than the t r a n s i t time o f the p r e s s u r e p u l s e through (A.5)  t h e p i e z o e l e c t r i c element.  holds only f o r i n f i n i t e l y  Equation  t h i n sensing elements.  Jones^^  has o b t a i n e d a r i s e - t i m e o f 0.54 ysec u s i n g a 10 cm b e r y l l i u m r o d ( s m a l l 1> , l a r g e c ) 5 mm i n d i a m e t e r p i e z o e l e c t r i c c e r a m i c element 0.5 mm  and a PZT-4  thick.  A p r e s s u r e probe must be c a r e f u l l y mounted i n o r d e r t o a l l o w f r e e motion o f the s u r f a c e o f t h e f r o n t r o d ^ ^ ' ^ . 4  F a i l u r e t o do so leads t o d i s t o r t e d s i g n a l s and l o n g r i s e A l s o t h e b a r must be p r o t e c t e d from l a t e r a l s t r e s s e s .  time.  131 We have c o n s t r u c t e d p r e s s u r e p r o b e s - u s i n g  fused q u a r t z  rods of 1 mm d i a m e t e r and PZT-4 ceramic d i s c s 0.2 mm  thick,  With f r o n t and r e a r rods 22 cm and 10 cm l o n g r e s p e c t i v e l y , t h e d e l a y t i m e , r i s e - t i m e , and o b s e r v a t i o n time were 40 y s e c , 0.8 y s e c , and 37 y s e c , r e s p e c t i v e l y .  The c o n s t r u c t i o n o f the  probe and i t s mount i s shown i n d e t a i l i n F i g u r e A-2 and F i g u r e 3-7.  The probe f i t t e d l o o s e l y i n s i d e the p l a s t i c t u b i n g  which was i n s i d e the Corex p r o t e c t i n g s h i e l d  ( k i n d l y donated  by  the C o r n i n g G l a s s Works). The procedure  f o r c o n s t r u c t i n g the probes was as f o l l o w s .  The q u a r t z rods were c u t t o l e n g t h and then t h e i r ends were c a r e f u l l y ground f l a t and p e r p e n d i c u l a r t o the axes o f the r o d s . The i n s u l a t i o n was removed from the ends o f #40 enameled w i r e s . Two o r t h r e e t u r n s o f the bare w i r e were wrapped around the rods near the ends which were t o be g l u e d t o the p i e z o e l e c t r i c  disc.  A t h i n f i l m o f c o n d u c t i n g p a i n t was then a p p l i e d t o the ends o f the rods and over the s m a l l c o i l s o f w i r e . still  While the p a i n t was  s o f t , the rods were mounted i n a s p e c i a l l y - m a d e v i s e , the  PZT-4 d i s c was i n s e r t e d between t h e ends of the two r o d s , and the rods were then clamped t i g h t l y a g a i n s t the d i s c .  When the  p a i n t had d r i e d , the d i s c and r o d ends were covered w i t h a s m a l l amount o f epoxy cement t o bond t h e p i e c e s f i r m l y t o g e t h e r . B e f o r e the glue hardened, the w i r e s were t w i s t e d t i g h t l y together. F i g u r e A-3 shows s i g n a l s o b t a i n e d w i t h these probes mounted p e r p e n d i c u l a r t o the a x i s o f the shock tube.  Brass  O-ring Seal  Tube  Covar Seal  Set-Screw  Brass  Brass  Tube  /  BNC Connector  f o r Adjusting  Probe  Twisted #40' Wire Leads  / 4I  i i  Soft Plastic Tube  Corex Tube  Figure  A-2  Construction Details  Front Quartz Rod  of Pressure Probe  Housing  PZT-4 Piezoelectric Ceramic Disc  / Rear* Quartz Rod  t  R e f l e c t i o n from Free End of Rear Rod (a)  Mach 9 Shock i n Argon a t 10 T o r r (100 mV/div, 5 u.sec/div)  (b)  Mach 13 Shock i n Argon a t 1 T o r r (10 mV/div, 2 j i s e c / d i v )  t "Sound" P u l s e ( i ) Without "Sound" (c)  ( i i ) With "Sound"  I l l u s t r a t i n g Use of P r e s s u r e Probe i n Sound Speed Measurements Behind Mach 13 Shock i n Argon a t 1 T o r r (10 mV/div, 2 u s e c / d i v )  F i g u r e A-3  T y p i c a l P r e s s u r e Probe S i g n a l s  

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